Embed Size (px)
Citation preview
LA-10363-MS
A Sample Problem for
Variance Reduction in MCNP
LosN A T I O N A L L A B O R A T O R Y
AlamosLos Alamos National Laboratory is operated by the University of Californiafor the United States Department of Energy under contract W-7405-ENG-36.
An Affirmative Action/Equal Opportunity Employer
This report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither The Regents of the University of California, the United States Governmentnor any agency thereof, nor any of their employees, makes any warranty, express or implied, orassumes any legal liability or responsibility for the accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, or represents that its use would not infringeprivately owned rights. Reference herein to any specific commercial product, process, or service bytrade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply itsendorsement, recommendation, or favoring by The Regents of the University of California, theUnited States Government, or any agency thereof. The views and opinions of authors expressedherein do not necessarily state or reflect those of The Regents of the University of California, theUnited States Government, or any agency thereof. The Los Alamos National Laboratory stronglysupports academic freedom and a researcher's right to publish; therefore, the Laboratory as aninstitution does not endorse the viewpoint of a publication or guarantee its technical correctness.
LA-10363-MS
LosAA
UC-32hued: October 1985
A Sample Problemfor Variance Reductionin IWICNP
Thomas E. Booth
IaliinmLosAlamosLosAlamos,
NationalLaboratoryNew Mexico87545
A SAMPLE PROBLEM FOR VARIANCE REDUCTION IN MCNP
by
ThomasE. Booth
ABSTRACT
The Los AlamoscomputercodeMonteCarloNeutronPhotonusefulvariancereductiontechniquesto aidthe MonteCarlouser.
(MCNP) has manyThis reportapplies
manyof these techniquesto a conceptuallysimple but computationallydemandingneutrontransportproblem.
I. INTRODUCTION
This report is based on a series of four 50-minvariance reduction talks (“MCNP Variance ReductionTechniques,”video reels#12-15)given at the MagneticFusion Energy Conference on MCNP,* Los AlamosNational Laboratory,October 1983.It is an overviewofall variance reduction techniques in MCNP and not anin-depthconsiderationofany. In fact, the techniquesaredescribed only in the context of a single conceptuallysimple, but demanding, neutron transport problem,with only enough theory presented to describe the gen-eral flavor of the techniques. Detailed descriptions arein the MCNP manual.1
This report assumesa generalfamiliarity with MonteCarlo transport vocabulary such as weight, roulette,score,bias, etc.
II. VARIANCEREDUCTION
Variance-reducingtechniques in Monte Carlo calcu-lationscan often reduce the computer time required toobtain resultsof sufficientprecision.Note that precision
*VideotapesoftheentireconferenceareavailablefromRadia-tionShieldingInformationCenter,OakRidgeNationalLabo-ratory,OakRidge,TN 37830.The readerwishingto run thesampleproblemhereshouldreferto the appendixbeginningonpage67forinputfiledetailsmodifiedsincetheconferenceandafterthewritingofthisreport.
is onlyone requirement for a goodMonte Carlo calcula-tion. Even a zero variance Monte Carlo calculationcannot accurately predict natural behavior if othersources of error are not minimized. Factors affectingaccuracywereoutlinedby Art Forster, LosAlamos(Fig.1).**
This paper demonstrates how variance reductiontechniquescan increasethe efficiencyof a Monte Carlocalculation.Two user choicesaffect that ei%ciency,thechoiceof tally and of random walk sampling.The tallychoice (of for example, collision vs track length esti-mators) amounts to trying to obtain the best resultsfrom the random walks sampled. The chosen randomwalk samplingamounts to preferentiallysampling“im-portant” particles at the expense of “unimportant”particles.
A. Figureof Merit
The measure of efficiencyfor MCNP calculations isthe figureof merit (FOM) defined as
1FOM = —
G:r T ‘
**Videoreel #l 1, “RelativeErrors,Figureof Merit” fromMCNPWorkshop,LosAlamosNationalLaboratory,October4-7, 1983.Availablefrom RadiationShieldingInformationCenter,Oak RidgeNational Laboratory,Oak Ridge,TN37830.
1
1. CODEFACTORS
PHYSICSANDMODELSDATAUNCER’PAINTIES
CROSS-SECTIONREPRESENTATION
ERRORSINTHECODING
2. PROBLEM-MODELINGFACTORS
SOURCEMODELANDDATAGEOMETRICALCONFIGURATIONMATERIALCOMPOSITION
3. USERFACTORS
USER-SUPPLIEDSUBROUTINEERRORS
INPUTERRORSvA~ANCEREDUcTIoNABUfjE
CHECKINGTHEOUTPUTUNDERSTANDINGTHEPHYSICALMEASUFtEMEIW
Fig. 1. Factorsaffectingaccuracy.
where am,= relative standard deviation of the mean andT = computer time for the calculation (in minutes). TheFOM should be roughly constant for a well-sampledproblembecauseO%ris (on average)proportional to N–l(N= number of histories) and T is (on average)propor-tional to N; therefore,the product remains approximatelyconstant.
B. GeneralComments
Although all variance reduction schemes have someunique features,a fewgeneralcomments are worthwhile.Considerthe problem of decreasing
(whereCJ2= history variance, N= number of particles,and p = mean) for fixed computer time T. To decreasea~,, we can try to decrease o or increase N—that is,decrease the time per particle history-or both. Un-fortunately, these two goals usually conflictbecause de-creasing a normally requires more time per history be-cause better information is required and increasing Nnormallyincreasesa becausethere is lesstime per historyto obtain information. However, the situation is nothopeless.It is often possible to decrease o substantially
without decreasingN too much or increase N substan-tiallywithout increasingo too much so that
ca“,,= —
Pm
decreases.Many techniques described here attempt to decrease
a~r by either producing or destroying particles. Sometechniquesdo both. In general, (1) techniques that pro-duce tracks work by decreasing a (we hope much fasterthan N decreases),and (2) techniques that destroy trackswork by increasing N (we hope much faster than aincreases).
IIIILTHE PROBLEM
The problem is illustrated in Fig. 3, but beforediscuss-ing its Monte Carlo aspects, I must point out that theproblem is atypical and not real. I invented the sampleproblem so most of the MCNP variance reduction tech-niquescould be applied. Usually, a real probiem will notneed so many techniques. Furthermore,without under-standing and caution, “variance-reducing” techniquesoften increasethe variance.
Figure2 is the input file for an analog MCNP calcula-tion and Fig. 3 is a slice through the geometry at z = O.
2
3
SAMPLE PROBLEM FOR MFE TALKSc TALLIES FOR PARTICLES WITH E>.OIMEV
i o (1 -21):-22 1 -2.03E0 -1 -3 23 1 -2.03E0 -1 -4 34
:789
1011121314151617181920
1 -2.03E0 -1 -5 41 -2.03E0 -1 -6i -2.03E0 -i -7 :1 -2.03E0 -1 -8 71 -2.03E0 -1 -9 81 -2.03E0 -1 -lo 91 -2.03E0 -1 -11 101 -2.03E0 -1 -12 111 -2.03E0 -1 -13 121 -2.03E0 -1 -~4 13i -2.03E0 -1 -15 141 -2.03E0 -1 -16 151 -2.03E0 -1 -17 16t -2.03E0 -1 -18 171 -2.03E0 -1 -19 18i -2.03E0 -t -20 190 -1 -21 20
21 1 -2.03E-2 -1 -22 21220 1 21 -2223022
:3456789
10111213141516171819202122
MODEcMl
SRCIS15PNPSINFIF4F5PooEOTOCLITNCTMEPROMPPRINT
CY 100PY oPY 10PY 20PY 30PY 40PY 50PY 60PY 70PY 80PY 90PY 100PY 110PY 120PY i30PY +40PY 150PY 160PY 170PY 180PY 2000PY 2010
0THE FOLLOWING IS
1001 -.010SCHAEFFER PORTLANO CONCRETE
8016 -.52911023 -.01612000 -.00213027 -.03414000 -.33719000 -.01320000 -.04426000 -.0146012 -.001
0 1.E-6 o 2 1.014 14
;25 .5 11000000 1 3R 15R2R02021200 2005 0 00 19R 1 0 0 ~ONLY CELL 21 CONTRIBUTES TO POINT.01 100100 1000 100001.0E123 0.0 0045-5 -5
DETECTOR TALLY
Fig. 2. InputfileforananalogMCNP calculation.
/CELL 21 CONCRETEp= 2.03E-2 ~/CC
F4 TRACK-LENGTHFLUX ESTIMATE
x
4
2000 cm
CONCRETEP= 2.Q3 91CC ~
e200
VOID
ELL 2(
~ 200+
4-—
POINT DETECTOR CONTRIBUTIONS
/ TAKEN ONLY FROM CELL.21
\>m
WANT FLUX HERE IF5 FLUX AT A POINT CELL22
VOID
CYLINDER 100-cm RADIUS. PARTICLES- CROSSING THIS SURFACE ARE “-.. . ‘-
(*PERFECT SHIELDS)
Fl (SURFACE CROSSING TALLY)
/3 CELL 2-19
POINT ISOTROPICNEUTRON SOURCE
25962 MeV50% 14 MeV
25% 2- 114MeV uniform
Fig. 3. Theproblem.
The primary tally is the point detector tally (F5)at the topof Fig.3, 200 cm from the axis of the cylinder(y-axis).Apoint isotropic neutron source is just barely inside thefirstcell(cell2)at the bottom of Fig.3.The sourceenergydistribution is 25%at 2 MeV, 50%at 14 MeV, and 25Y0uniformly distributed between 2 and 14 MeV. For thisproblem, the detectors will respond only to neutronsabove0.01 MeV.
A“perfectshield”immediatelykillsany neutrons leav-ing the cylinder(exceptfrom cell 21 to cell 22). Thus, totally (F5),a neutron must
1. penetrate 180cm of concrete(cells2-19),2. leave the concrete (cell 19) with a direction close
enough to the cylinder axis that the neutron goesfrom the bottom of cell 20 (the cylindricalvoid) tothe top and crossesinto cell21,
3. collide in cell 21 (becausepoint detector contribu-tions are made only from collision/sourcepoints),and
4. have energyabove 0.01 MeV.These eventsare unlikelybecause
1.
2.
3.
4.
180 cm of 2.03-g/cm3 concrete is difficult topenetrate,there is only a small solid angle up the “pipe” (cell20),not many collisionswill occur in 10 cm of 0.0203-g/cm3concrete,andparticlesloseenergypenetrating the concrete.
Before approaching these four problems, knowledgeabout the the point detector technique can be applied tokeep from wasting time; only collisions in cell 21 cancontribute to the point detector. Collisionsin cells 2-19cannot contribute through the perfect shield, that is, zeroimportance region. Thus, the MCNP input is set (PDOcard, Fig. 2) so that the point detector ignorescollisionsnot in cell 21. If the point detector did not ignore col-lisionsin cells2-19, the followingwould happen at eachcollision.
1.
2.
3.
4.
The probabilitydensity for scattering toward thepoint detectorwould be calculated.A point detector pseudoparticle would be createdand pointed toward the point detector.The pseudoparticlewould be tracked and exponen-tiallyattenuated through the concrete.The useudoparticle would eventually enter theperfe&shield(cell1)and be killedbeca&e a straightline from any point in cells 2-19 to the point de-tector would enter the perfectshield.
There is no point proceedingwith these stepsbecausethepseudoparticlesfrom cells2-19 are alwayskilled;time issaved by ignoringpoint detector contributions from cells2-19.
IV. ANALOGCALCULATION
Inspection of Fig. 4, which is derived from MCNPsummary tables, shows that the analog calculation fails.Note that the tracksenteringdwindleto zero as they try topenetrate the concrete (cells2-19).This problem will beaddressed in more detail later, but first note that thenumber weightedenergy(NWE)is very low,especiallyincells 12, 13, and 14. The NWE is simply the averageenergy,that isNWE= JN(E)E dE
~N(E) dE ‘
where E = energy and N(E) = number density at energyE.This indicatesthat there are many neutrons below0.01MeVthat the point detector willnot respond to. There isno sense followingparticles too low in energy to con-tribute; therefore, MCNP kills neutrons when they fallbelowa user-suppliedenergycutoff.
V. ENERGY AND TIME CUTOFFS
A. EnergyCutoff
The energycutoff in MCNP is a singleuser-suppliedproblem-wide energy level. Particles are terminatedwhen their energy falls below the energy cutoff. Theenergy cutoff terminates tracks and thus decreases thetime per history.The energycutoff should be used onlywhen it is knownthat low-energyparticlesare either ofzero importance or almost zero importance. A numberof uitfallsexist.
i.
2.
3.
Remember that low-energyparticlescan often pro-duce high-energyparticles (for example, fission orlow-energy neutrons inducing high-energyphotons).Thus, even if a detector is not sensitiveto low-energyparticles, the low-energy particlesmay be important to the tally.The energy cutoff is the same throughout theentire problem. Often low-energyparticles havezero importance in some regions and high im-portance in others.The answerwillbe biased(low)if the energycutoffis killingparticles that might otherwisehave con-tributed. Furthermore, as N+IXI the apparent er-ror will go to zero and therefore mislead the un-wary.Seriousconsiderationshouldbe givento twotechniques (discussed later), energy roulette andspace-energyweight window, that are always un-biased.
5
CELL
PROGR PROBL
2 23 34 45 56 67 78 8~ ~
10 1011 1112 12!3 !3f4 f415 1516 16i7 1718 1819 1920 2021 2122 22
TOTAL
TRACKSENTERING
4783217615639395112871708744313118
:o000000
10644
POPULATION COLL:
3931931593362205115
63401610
76200000000
6281
SIONS COLLISIONS* WEIGHT
(PER HISTORY)
139491505712510
7390421322191587
96130423033021e
1700000000
58985
3.5593E+O03.8421E+O03.1921E+O01.8857E+O01.0750E+O05.6622E-014.0495E-01z.45z2E-oj7.7571E-025.8688E-028.4205E-025.5626E-024.3378E-03o.
::o.0.0.0.0.
1.5051E+OI
NUMBERWEIGHTED
ENERGY
2.4144E-034.5943E-042.0566E-041.4450E-049.3995E-051.0022E-046.4696E-056.2827E-051.0691E-046.2272E-052.2207E-051.993+E-063.7686E-06o.0.0.0.0.0.0.0.
FLUXWEIGHTEO
ENERGY
4.7075E+O01.9643E+O01.2067E+O08.4454E-015.6654E-015.8205E-014.4866E-014.6476E-Oi5.1448E-012.4500E-011.1767E-016.6168E-047.9961E-04o.0.0.0.0.0.0.0.
AVERAGETRACK WEIGHT
(RELATIVE)
I.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OO1.0000E+OOI.0000E+OOI.0000E+OO1.0000E+OOt.0000E+OOI.0000E+OOo.0.0.0.0.0.0.0.
AVERAGETRACK MFP
(CM)
5.8207E+O03.9404E+o03.3058E+O03.0062E+O02.7411E+O02.7733E+O02.5496E+O02.6046E+O03.0390E+O02.4143E+O02.t16tE+O01.919~E+O02.0823E+O0o.0.0.0.0.0.0.0.
ANALOG CALCULATION - NO VARIANCE REDUCTION TECHNIQUES
TALLY I TALLY 4NPS MEAN ERROR I=OM MEAN ERROR Fokl1000 0. 0.0000 0 0. 0.0000 02000 0. 0.0000 0 0. 0.0000 03000 0. 0.0000 0 0. 0.0000 03919 0. 0.000o 0 0. 0.0000 0
TALLY 5MEAN ERROR FOM
o. 0.0000 00. 0.0000 00. 0.0000 00. 0.0000 0
******%*******************************************************************OUMP NO. 2 ON” FILE RUNTPE NPS = 3919 CTM = .61
Fig.4. Analogcalculation.
B. TimeCutoff
The time cutoff in MCNP is a single user-supplied,problem-widetime value.Particlesare terminated whentheir time exceeds the time cutoff. The time cutoffterminates tracks and thus decreasesthe computer timeper history.The time cutoffshouldonlybe used in time-dependent problems where the last time bin will beearlierthan the cutoff.
The sample problem in this report is time-independ-ent, so the time cutoff is not demonstrated here.
C. TheSampleProblemwithEnergyCutoff
Figure 5 gives the results of an MCNP calculationwith a O.01-MeVenergy cutoff. Note that the numberweighted energy is about 1000 times higher, so theenergy cutoff has changed the energy spectrum as ex-pected. Furthermore, note that about four times asmany historieswere run in the same time although thetotal number of collisionsis approximatelyconstant.
Despite more histories, fewer tracks enter deep intothe concrete cylinder. This may seem a little counter-intuitive until one remembers that the energy cutoffkills the typical particle that has had many collisionsand is below the energy cutoff, that is, the typicalparticledeep in the concrete.This decreasein the tracksentering is not alarming because we know that onlytracks with energy less than 0.01 MeV were killed andthey cannot tally.
The trouble with the calculation is that the largeamount of concrete is preventing neutron travel fromthe sourceto the tallyregion.The solutionis to preferen-tiallypush particlesup the cylinder.Four techniques inMCNP can be used for penetration,
1. geometrysplitting/Russianroulette,2. exponentialtransorm,3. forcedcollisions,*and4. weightwindow.
VI. GEOMETRY SPLITTING ANDRUSSIAN ROULETTE
Geometry splitting/Russian roulette is one of theoldest,most widelyused variance reduction techniques.As with most biasing techniques, the objective is tospend more time samplingimportant (spatial)cellsandless time sampling unimportant cells. The technique(Fig.6) is to
1. divide the geometryinto cells;2. assignimportances (In)to these cells;and
*Therewill not be an exampleusingforcedcollisionsforpenetrationproblemsbecauseit isawkwardtodoinMCNP.Infact,analterationto theweightcutoffgameisoftennecessary.
3. when crossing from cell m to cell n, computeV = In/l~. Ifa. v = 1,continue transport;b. v <1, play Russian roulette,c. v >1, split the particle into v = I./I~ tracks.
A. RussianRoulette(v < 1)
If v <1, the particle is entering a cell that we wish tosample less frequently, so the particle plays Russianroulette.That is,
1. with probability v, the particle survives and itsweightis multipliedby V–l,or
2. with probability 1– v the particle is killed.In general, Russian roulette increases the historyvariance but decreases the time per history, allowingmore historiesto be run.
B. Splitting(v> 1)
If v > 1, the particle is entering a more importantregionand is split into “v” subparticles.Ifv is an integer,this is easy to do; otherwise v must be sampled. Con-sidern < v < n + 1,then
Probability SplitWeight
p(n) = n + 1– v wt, = wt/n sampledp(n+l) = v – n wt, = wt/(n + 1) splitting
The sampled splittingschemeabove conservesthe totalweightcrossingthe splittingsurface,but the split weightvaries, depending on whether n or n + 1 particles areselected.
L Actually,MCNP does not use the sampled splittingscheme.MCNP usesan expectedvalue scheme:
Probability SplitWeight
p(n) = n + 1– v wt, = wt/v expectedvaluep(n + 1)= v – n v-n,= wt/v splitting
The MCNP schemedoes not conserveweightcrossingasplittingsurfaceat each occurrence.That is, if n particlesare sampled,the total weightentering is
wn. — =~. wt<wt,
Vv
but if n + 1 particles are sampled, the total weightenteringis
(n+ 1) ~t = %1 wt> wt,
However,the expectedweightcrossingthe surfaceis wt:Wt
p(n)” n” ~ + p(n + 1)”(n + 1)” ~= wt.
7
8
CELL
PROGR PR08L
2 23 34 45 56 67 78 89 9
10 1011 1112 1213 1314 1415 1516 16i7 171819 ;:20 2021 2122 22
TOTAL
NPS
1%3om4000500060007cummoo9000
100DO1100012000130ecii391m
TRACKS POPULATIONENTERING
15416 140044445 30982f97 1580
973 716467 331233 17j110 6556 4340 2420 15
8 73 200 :o 00 00 00 00 00 00, 0
23968 20076
TALLY tHEAN
::o.0.0.0.0.0.0.
::o.0.0.
ERROR0:0000O.0000
:%%O.00000.00000:0000o.OoooO.ocmoO.0000
::%%
%%%
FOMo0
:o000000000
27380156117830366117267654201861557842
:,00000000
57662
\
1.9802E+O01. I176E+O05.6057E-012.6210E-Oi1.2357E-015.4768E-023.0069E-021.3316E-021.i097E-025.5842E-033.0069E-035.7274E-04o.0.0.0.0.0.0.0.0.
\4.1425E+o0
NUMBERWEIGHTEO
ENERGY
2.2661E+O01.0718E+o08.8425E-017.9762E-017.2799E-018.0i05E-017.4618E-Ot9.0855E-015.8161E-015.31OOE-OI4.0663E-014.8527E-02o.0.0.0.0.0.0.0. \
FLUXWEIGHTEO
ENERGY
6.0L130E+o03.96S8E+O03.4412E+O02.9569E+O02.6838E+O02.6783E+O0
AVERAGETRAcK WEIGHT
(RELATIVE)
I.0000E+OOI.0000E+OOI.0000E+OOI.0000E+ooI.0000E+ooI.0000E+OO
2.4966E+002.6749E+O01.7610E+O0i.6936E+001.5i99E+O03.3019E-01o.0.0.0.0.0.0.0. I
1.00CK3E+O0I.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOo.0.0.0.0.0.0.0.
0
“(o. 0.
COLLISIONS PER HISTORYHAS DECREASEO
AVERAGE ENERQYHAS INCREASISD
COLLISIONS COLLISIONS* WEIGHT
(pER HISTORY)
********e****.******.*********.*******.*.******.****************Q6***o****OUMP No. 2 ON FILE RUNTPF NPs = 13968 cm! * .6o
NOTES:
1) N INCREASED FROM 3919 TO 13968
ENERGY CUTOFF-O.01 MeV 2) TRACKS STOP SOONER BECAIJSE OF ENERGY CUTOFF
31 PARTICLES NOT GETTING TO TALLY REGIONS
AVERAGETRACK MFP
(CM)
6.9886E+O05.9464E+O05.6866E+O05.5260E+O05.4005E+O05.5811E+O05.5021E+O05.6290E+O04.9657E+O04.4423E+O04.6999E+O03.1147E+O0o.0.0.0.0.0.0.0.0. \
HAS INCREASED
LTOTAL NUMBER oFcoLLleloNs pRocESsEDABOUTTHESAME
TALLYMEAN
o.0.0.0.0.0.::o.0.0.0.0.0.
4ERROR
O.00000.00000.0000O.omo0.00CQO.0000
::%%
::%%o.OlxloO.olx)o0.000oO.ocm
FOMo0000000000000
TALLY 5MEAN
o.0.0.0.::o.0.0.0.0.0.0.0.
ERROR FOMO.0000 0O.moo oO.ofxlo o0.0000O.owo :O.0000 00.0000 00.0000 00.0000 0O.ocmo o0.0CKM3 o
0::%%0.000o :
Fig,5. EnergycutoffofO.01MeV.
I = IMPORTANCE EXAMPLE T2LE = 3j
/
o
+
SPLITTING113
i 1f3113
11- 113or
xKILL t
113
RUSSIAFlROULETTE
I\
\
14
ISPLITTING SURFACES
Fig.6. Geometrysplitting/Russianroulettetechnique.
T
The MCNP scheme has the advantage that all parti-cles crossing the surface will have weight wt/v.Furthermore, if
1. geometry splitting/Russian roulette is the onlynonanalogtechnique used and
2. all source particles start in a cell of importance ISwith weightw,, then all particlesin cellj willhaveweight
L
regardlessof the random walktaken to cellj.MCNP’S geometry splitting/Russian roulette in-
troducesno variancein particleweightwithin a cell.Thevariation in the rzwnberof tracks scoringrather than avariation in particle weight determines the historyvariance. Empirically, it has been shown that largevariations in particleweightsaffecttalliesdeleteriously.Booth2 has shown theoretically that expected valuesplitting is superior to sampled splitting in high-variancesituations.
C. CommentsonGeometrySplitting/RussianRoulette
One other small facet deserves mention. MCNPnever splits into a void although Russian roulette maybe played entering a void. Splitting into a void ac-complishesnothingexceptextra trackingbecauseall thesplit particlesmust be tracked across the void and they
10
all make it to the next surflace.The stdit should be done.according to the importance ratio of the last nonvoidcelldeparted and the first nonvoid cell entered (integersplittinginto avoid wastestime, but it does not increasethe history variance). In contrast, noninteger splittinginto a void may increasethe history variance and wastetime.
Finally, splitting generally decreases the historyvariancebut increasesthe time per history.
Note three more items:1.
2.
3.
Geometry splitting/Russian roulette works wellonly in problems without extreme angular de-pendence. In the extreme case, splitting/Russianroulettecan be uselessif no particlesever enter animportant cellwhere the particlescan be split.Geometry splitting/Russianroulette will preserveweightvariations.The technique is “dumb” in thesense that it never looks at the particle weightbeforedecidingappropriate action. An example isgeometry splitting/Russian roulette used withsourcebiasing.Geometry splitting/Russianroulette are turned onor off together.
D. Cautions
Althoughsplitting/Russianroulette is among the old-est, easiest to use, and most effective techniques inMCNP, it can be abused. Two common abusesare:
9
1. compensating for previous poor sampling by avery largeimportance ratio and doing the splitting“all at once.”
2<
‘XAM’LE’S=I’I2I 4i ‘ H321S::::NG
SOURCE
-1 I I1 1 8
using splitting/Russian roulette with other tech-niques (for example, exponential transform)without forethought to possible interference ef-fects.
E. TheSampleProblemwithGeometrySplitting/RussianRoulette
Returning to the problem, recall
Cell TracksProgr Probl Entering
SourceCell 23456789
10111213141516171819202122
2345,6789
10111213141516171819202122
1541644452197973467233110564020
83000000000
Note that except for the source cell, the tracks enteringare decreasingby about a factor of 2 in each subsequent
ill
BAD8 8 32 SPLITTING
cell.Furthermore, becausehalf the particlesfrom cell 2(the sourcecell)immediatelyexit the geometryfrom theisotropicsource,the rough factorof 2 even holds for thesource cell.Thus as a first rough guess, try importanceratios of 2:1 through the concrete; that is, factor of 2splitting.
Figure 7 indicates that this splitting is much betterthan no splitting.Not onlydid particlesfinallypenetratethe concrete (see Tally 1) but the “tracks entering”column is roughlyconstant within a factor of 2. Slightlymore,splitting in cells 9-19 might improve the “tracks ‘entering”just a little bit more. The splittingratios wererefined to be 2 in cells 2-8 and 2.15 in cells 9-19 in thenextcalculation.
Figure 8 summarizes the refined splitting. Im-mediately evident is that the FOM (Tally 1) unex-pectedly decreased tlom 27 (Fig. 7) to 23, so at firstglance, the refined splitting appears worse. However,note that the refined splittinghad the desired effect;the“tracks entering” numbers are flatter. Thus I think therefinedsplittingis better despite the lowerFOM.
What justifies being so cavalier about FOMS? Re-memberthat the FOM is onlyan estimate of the calcula-tional efilciency.At relative-error estimates near 25Y0,these FOMSare not meaningfulenough to take the 27-to-23FOM differenceseriously.Furthermore, the FOMis only one of the many available pieces of summaryinformation. At 25%error levels, it is much more im-portant that the refined splittingappears to be samplingthe geometrybetter.
F. Discussionof Results
The effectof refined’splittingin this ,sampleproblemillustratesan important point about most variance re-duction techniques; most of the improvement canusually be gained on the first try. Either one of thesesplitting/Russianroulette runs is severalorders of mag-nitude better than the run without splitting.IrIfact, this
10
NOTE:MUCHBETTER 7
CELL TRACKS\ENTERING
PROGR PROBL
2 23 34 45 56 67 78 89 9
10 1011 1112 1213 1314 *415 1516 1617 1718 18
y:-
13231321132613531358126111821089998823792734664525514
POPULATION
1 222911191140113111561154117710811013931853697678623568453441375163
COLLISIONS COLLISIONS* WEIGHT
4oi 144244803471946875038488147774632411837553127322027982489f80718261323
r
o00
66435
(PER HISTORY)
1. 8938E+O01. 0444E+O05. 6693E-012.7851 E-011.3831 E-017. 4333E-023. 6008E -021.7621 E-028. 5428E -033.7974E-031.7313E-037.2089E-043.71i7E-041.6126E-047.1726E-052.6036E-051.3155E-054.7657E-06o.0.0.
4.0653E+O0
NUMBERWEIGHTEO
ENERGY
2.2793E+O01.1595E+O09.6248E-018.8914E-018.3666E-017.i902E-Oi7.5760E-016.7145E-016.2347E-015.7569E-015.5764E-015.8678E-015.5526E-015.5983E-015.6580E-016.1889E-015.7871E-017.4292E-011.5219E+O0
::
FLUXWEIGHTEO
ENERGY
6.0872E+O04.2228E+O03.5562E+O03.1819E+O02.9173E+O02.5852E+O02.6075E+O02.2906E+O02. I041E+O01.9068E+O01.7904E+O01.80i5E+O01.7102E+O01.7822E+O01.8696E+O0i.9954E+oo1.8610E+O02.19f8E+O03.3568E+O0o.0.
AVERAGETRACK WEIGHT
(RELATIVE)
I.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOi.0000E+OOI.0000E+OOI.OUOOE+OOI.0000E+OOf.000oE+ooI.0000E+OO$.0000E+OOI.0000E+OOI.0000E+OOi.000oE+ooI.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOo.0.
AVERAGETRACK MFP
(CM)
7.0075E+O06.0880E+O05.8928E+O05.7399E+O05.6134E+O05.3490E+O05.4513E+o05.2870E+O05.2385E+O05.1391E+o05.0335E+o05.1663E+O05.0461E+O05.1226E+O05.1708E+O05.2508E+O05.1414E+O05.5246E+O01.0000+123o.0.
NEXT TO A VOID
\
VOID CELL
L F4 TALLY IN THIS TOP CELL IS ZERO BECAUSE NO Particles EVER ENTERED THE CELL
NEXT RUN : INCREASE SPLITTING AT CELL 9 TO 2.15 UNTIL CELL 19
TALLY 1 TALLY 4 TALLY 5NPS MEAN ERROR FOM MEAN ERROR FOM MEAN ERROR FOM1000 8.08716E-07 .3220 31 0. 0.0000 0 0.
WE GOT A TALLYI0.0000 0
~moo 5.95093E-07 .2532 27 0. 0.0000 0 0. 0.0000 02118 5.87154E-07 .2445 27 0. 0.000o 0 0. 0.0000 0
**************************************************************************DUMP NO. 2 ON FILE RUNTPG NPS = 2118 CTM = .60
\ PARTICLES STARTED HAVEDECREASED FROM 13968
I-P Fig.7. Factorof2spMtingfiorncefls2to19.
=00
0ILwoo
000
IZO
Ow
oow
..00
m
z
>ti
AZ
-1<1-
00
+r-W
aw
u->
au
-lcY
3F-
W>X5
avwarx
cc-k
0000000000000000000000000000000000000000+
++
+-+
++
++
++
++
++
++
++
+Lu
uu
llum
uu
lu.!u
uu
uu
ullu
uu
ulu
u00000000000000000000
00000000000000000
?0008000000.000
,58Z
%2Z
%%
80000000000
.r.......--r-.-.....o
Im(5zi=1-inu)u
)
oo~.-mci~m
mm
e~vmm
mm
000000000000000000+
+11s1111s
IlltB
Blt
Ww
ululmw
luww
wulluw
ww
wulw
m-m
incnw-m
-ww
mm
mm
m~m
rkw
mw
mow
mm
m=tw
wm
mm
mm
tncn
twqw
r.omm
mw
+-b
mw
owm
otnomm
wm
oo-m.-m
mm
mm
o0Lommd’wIll
N-tw
-1o*fYU
wa
***Iml
*II
**m
=w
m;2
o@4@
4
“(’******/
z0*L
U*L*l-*2
!*2*!X**IU
:d*IL**Z*O**N*
\mlx0
-1nco
*
12
problem is so bad without splitting that it is hard toguess how much splitting/Russian roulette has im-proved the efficiency.Contrast this improvement to the(questionable)FOM differenceof 27 (Fig.7) to 23 (Fig.8) between the factor of 2 splitting and the refinedsplitting. Usually one can do better with a variancereduction techniqueon the second try than on the first,but usuallyby not more than a factor of 2.
Quicklyreachingdiminishingreturns is characteristicof a competent user and a good variance reductiontechnique. Competent users can quickly learn goodimportancesbecausethere is a very broad near-optimalrange.Becausethe optimum isbroad, the statisticsoftenmask which importance set is best when they are all inthe vicinityof the optimum.
Now that a reasonablyflat track distribution has beenobtained, perhaps it is time to explain why one expectsthis to be near optimal. There are some plausibleargu-ments, but the real reason is empirical; it has beenobserved in many similar problems (that is, essentiallyone-dimensionalbulk penetration problems) that a flattrack distribution is near optimal. The radius of theconcrete cylinder is large enough (100 cm) that thecylinder appears much like a slab; very few particlescross its cylindrical surface at a given depth (c-oordinate) compared to the particle population at thatdepth. Indeed, if the radius were infinite, the cylinderwould be a slab and no particles would cross its cylin-drical surface.
Aplausibleargument for flat track distribution can bemade by considering an extremely thick slab andpossibletrack distributions for two cases. For too littlesplitting, the track population will decrease roughlyexponentially with increasing depth and no particleswill ever penetrate the slab. For too much splitting,theimportanceratiosare too large;the track populationwillincrease roughly exponentially and a particle historywill never terminate. In both cases, albeit for differentreasons, there are never any tallies. If neither an ex-ponentiallydecreasingpopulation nor an exponentiallyincreasingpopulation is advisable, the only choice is aflatdistribution.
Of course, there are really many more choices thanexponentiallydecreasing,flat, or exponentiallyincreas-ingpopulations,but track populationsusuallybehave inone of these ways because the importance ratios fromone cell to the next are normally chosen (at least for afirst guess) equal. The reason is that one cell in theinterior is essentiallyequivalent to the next cell,so thereis littlebasisto choosea differentimportance ratio fromone cell to the next. However, the cells are not quiteequivalent because they are different depths from thesource, so the average energy (and mean free path)decreaseswith increasingdepth. This is probablywhy itwasnecessaryto increasethe importance ratio from 2 to
2.15 in the deep parts of the sample problem. Note,however,that this is a small correction.
Returning to Fig. 8, note that the energy and meanfree path decrease with increasing depth, as expected.Not also that the higher splitting has decreased theparticlesper minute.
VII. ENERGYSPLITTING/ROULETTE
Energy splitting/Russian roulette is very similar togeometrysplitting/Russianroulette exceptenergysplit-ting/rouletteisdone in the energydomain rather than inthe spatialdomain. Note two differences.
1. Unlike geometry splitting/roulette, the energysplitting/rouletteusesactual splittingratios as sup-plied in the input file rather than obtaining theratios from importances.
2. It is possibleto play energysplitting/rouletteonlyon energydecreasesif desired.
There are two cautions.1. The weightcutoff game takes no account of what
has occurredwith energysplitting/roulette.2. Energysplitting/rouletteis played throughout the
entire problem. Consider using a space-energyweightwindowif there is a substantialspacevaria-tion in what energiesare important.
One can expect an improvement in speed usingenergyrouletteby recallingthat the problemran a factorof 4 faster with an energy cutoff of 0.01 MeV thanwithout an energy cutoff. Low-energy particles getprogressivelyless important as their energydrops, so itmight help to play Russian roulette at several differentenergiesas the energydrops. In the followingrun, a 50!40survivalgame was played at 5 MeV, 1 MeV, 0.3 MeV,0.1 MeV, and 0.03 MeV. The energies and the 50Y0survivalprobabilitywere onlyguesses.
The energy roulette (splitting does not happen herebecausethere isno upscatter)resultsare shownin Fig.9.Note that there were substantially(-50%) more tracksentering,approximatelythe same number of collisions,and three times as many particles run. The FOM looksbetter, but the mean (Tally 1)has increasedfrom 5.OE-7(Fig. 8) to 8.4E-7.This deserves note and caution, butnot panic,becausethe error is 18°%1,so poor estimates inboth tally and error can be expected.Despite the previ-ous statement, the energy roulette looks successful inimprovingtallies 1and 4.
VIII. IMPLICIT CAPTURE AND WEIGHTCUTOFF
A. ImplicitCapture
Implicit capture, survival biasing,and absorption byweightreduction are synonymous.Implicit capture is a
13
OO
OO
OO
OC
9C
NO
OO
OO
OO
NO
+++++++-+
u.!u
lluu
IILlu
JLu
+iu
To!n
maou
loul
.-.-~g:?goo
WF.N-NNU132
=00000
i?.
..
..
..
..
.InlnlnullnL
olnT=
ko
1-
LIJz;
u->
<U
J1-1
Cizc+
UJ>X2
<vuaufz-
1-
0000000000000000000000000000000000000000+
+-+
++
++
++
++
++
++
++
++
+U
IJ.IU
JU
JW
UJ
UIU
U!L
UU
IUJ
U!U
JU
.JU
IUJ
UU
IUJ
Fcx3
N1.@
Jm
ocom-3
Ln
wt-o)!aw
wow
alo~
om=fa
--=tm
wm
omm
mln
mao
a)o-lmm
=tn
ot.wom
!nm
.c+m
tiw~o
elwm
m~.cx-=
$w
m=tm
wbw
win
mo
..
..
.,.,.,,
..
..
..
..
..
-yv-mm
~~flm
m~~~m
m~w
q~o
aooooo
;i?gggg
u!
..
..
.00000
In
zu
ya1-
00000
000000
$)C
Jyclfy
UJ
ILJ
UIU
UJ
UJ
u’m
twm
ow+C
omt-vcn
N-ln
ulot.
‘r)oa-)ow
-..
.,,...-m
-N
N-o
u)
k-.-.
..
WC
9C
9C
9<
c+@
J@
JN
NN
NC
NC
-JN
**-*W44.
*****II
I=*i-
*U********s
~~zz5zzGzzzEzzzGzz8zi
++8111181
111111111+
IuwuJwwlAlwuwwwlLlulwululwuluJw
IDcOm.smmlD-a30srlnooomw.m@4N
~mr9mr9bomme4mwcn-f.o-mww-
mm
mr.t-m
pm
wm
n-t-w
oti-mow
c900c9
a+m
m+N
lD-c9
cJw
cQm
voN.
..
..
..
..
..
..
..
..
..
..
mtv
coca----
.,,,,..,,,.
,..,,,.
..
.--m
~-m
m-b
~-~
mv~
m.w
oooi*m*m*w**\.
JLn
zol-l(nH-1-10u
ao
Ui
+-
*+w
mom
+w
*oom
mm
mrzivm
-cot-it
—u
mc+
Nr-
*W
...,.*
O.m
aowm
wm
wm
otwm
mm
cioowo
wtim
m+w
mm
mt-rJ
Om
-ww
m*b
mm
w*N
-o--titwc-4
-c4c4
t-4c4
t-4m
*.--.--.------TF
-T---
U3Z
x-lV
wauLYl-
I-Zu
l
JmoCt
n.
w(3oCY
n.
-1-1uu
14
vaiiance reduction technique applied in MCNP afierthe collisionnuclidehas been selected.Let
~li= total microscopiccross sectionfor nuclide i and~.i = microscopicabsorptioncrosssectionfornuclide
i.When implicit capture is used rather than samplingforabsorption with probability ~~i/~~i,the particle alwayssurvivesthe collisionand is followedwith new weight
() CY~iWt “ 1— — .
Gti
Two advantagesof implicit capture are1.
2.
a particle that has finally, against considerableodds, reached the tally region is not absorbedjustbeforea tally is made, andthe history variance, in general, decreases whenthe survivingweight(that is, Oor W) is not sam-pled, but an expected surviving weight is usedinstead (but see weight cutoff discussion, Sec.VIII.B).
Two disadvantagesare1. implicit capture introduces fluctuation in particle
weightand2. increases the time per history (but see weight
cutoffdiscussion,Sec.VIII.B).Note that
1.
2.
3.
4.
Implicit capture is the default in MCNP (exceptfor note 4).Implicit capture is alwaysturned on for neutronsunlessthe weightcutoffgame is turned off.Explicit (analog) capture is not allowed for thephoton simplephysicstreatment (highenergy).Analogcapture is allowedonly in detailed photonphysics.
B. WeightCutoff
In weightcutoff, Russian roulette is played if a par-ticle’sweightdropsbelowa user-specifiedweightcutoff.The particleis either killedor its weightis increasedto auser-specifiedlevel. The weight cutoff was originallyenvisioned for use with geometry splitting/Russianrouletteand implicit capture. Becauseof this.
1.
2.
the weight-cutoffsin cellj depend not onlyon WC1and WC2 (see Fig. 2) on the CUTN and CUTPcards, but also on the cell importances. This de-pendence is intended to adjust the weight cutoffvalues to make sense with geometry split-ting/Russian roulette.Implicit capture is always turned on (except indetailed photon physics) whenever a nonzeroWC1 is specified.
The weight cutoffs WC1 and WC2 are illustrated inFig. 10. If a particle’s weight falls below Rj“WC2, a
weight cutoff game is played; with probabilitywt/(WCl oRj) the particle survives with new weightWC1 cRj;otherwisethe particle is killed.
As mentioned earlier, the weight cutoff game wasoriginally envisioned for use with geometry splittingand implicit capture. Consider what can happenwithout a weight cutoff. Suppose a particle is in theintenor of a very large medium and there are no timenor energycutoffs.The particlewillgofrom collisiontocollision,losinga fraction of its weightat each collision.Without a weight cutoff, the particle’s weight wouldeventuallybe too small to be representablein the com-puter, at which time an error would occur. If there areother loss mechanisms (for example, escape, timecutoff, or energy cutoff), the particle’sweight will “notdecrease indefinitely,but the particle may take an un-duly long time to terminate.
Weightcutoffs dependence on the importance ratiocan be easily understood if one remembers that theweightcutoffgame was originallydesignedto solve thelow-weightproblem sometimes produced by implicitcapture. In a high-importance region, the weights arelow by design, so it makes no sense to play the sameweightcutoffgamein high-and low-importanceregions.In fact,as mentioned in a previoussection,if splittingisthe only nonanalog technique used, all particles in agiven cell have the same weight, so no weight cutoffgamewould make sense.That is, if the particleweightistoo small in a cell, the cell importance simply needs tobe decreased. The weight cutoff is meant to indicatewhena particle’sweightis too lowto be worth transport-ing.
In addition to the weightcutoffs dependenceon cellimportance, the weightcutoffs are automatically maderelativeto the minimum sourceweightif the sourceis astandard MCNP source and the weight cutoffs (WC1,WC2)are prefixedby a negativesign.
1.a.
b.
2.a.
CautionsMany techniques in MCNP cause weightchange;the weight cutoff was really designed withgeometry splitting and implicit capture in mind.Care should be taken in the use of other tech-niques.In most cases, if you specifya weightcutoff, youautomaticallyget implicit capture.
NotesWeight cutoff games are unlike time and energycutoffs. In time and energy cutoffs, the randomwalk is always terminated when the threshold iscrossed. Potential bias may result if the particle’simportance was not zero. A weightcutoff (weightroulette would be a better name) does not bias thegame because the weight is increased for thoseparticlesthat survive.
15
-.1-Wc 1 “ Rj -— WEIGHT FOR PARTICLES
az SURVIVING THE WEIGHT$+ CUTOFF GAME+u-Elu — WC2 “ Rj 4—g~
SMALLEST ALLOWEDWEIGHT IN CELL, j
o
1 = CELL
Fig. 10. Weightcutoffmechanics.
b. T3ydefault,theweightcutoff gameisturned offinaweightwindowcell.
C. WeightCutoffand ImplicitCaptureAppliedto theSampleProblem
Figure 11showsthe resultofaddingweightcutoffandimplicitcapture techniquesin addition to the
1. energycutoff,2. refinedgeometrysplitting/Russianroulette, and3. energyroulette techniques.
Comparing Fig. 11 to Fig. 9, one can see that implicitcaptureand weightcutoffdid apparentlyreducethe tally1 error for the same number of particles.However, thenumber of particles run was down by a factor of 2,resultingin a net decrease in the FOM. In general, if a,nonanalogtechnique does not show a clear improve-ment, do not use it; thus for the next run, the implicitcapture and weightcutoffwillbe turned off.
Tally 1seemsreasonablywelloptimizedby1. geometrysplittingand roulette,2. energycuto~3. energyroulette (and splitting),and4. analogcapture.
Tally 4 is bad becausevery few tracks exit the concretecylinder (cell 19)in the small solid angle subtended bycell 21. Tally 5 is even worse, in fact nonexistent,becauseof the few particles that do reach cell 21, nonecollide,so there are no point detector contributions.
Considerimprovingthe worst tally(tally5)first.Notefrom the summary charts that the free path in cell 21 is-1000 cm and the cell is ‘1Ocm thick. Only a tinyfractionof the particlesenteringcell21 willcollidein ananalog fashion. The forced collision technique inMCNP solves this problem by requiring each trackenteringa cellto collide.
IMPORTANCE
IX. FORCEDCOLLISIONS
Forcedcollisionis normallyused to samplecollisionsin opticallythin (fractionalmean free path) cellswherenot enoughcollisionsare being sampled. A track enter-ing a forced collision cell is split into two tracks: un-collided and collided. That is, MCNP calculates theexpected weight traversing the cell and assigns thatweightto the uncollidedtrack,and MCNP calculatestheexpected weight colliding in the cell and assigns thatweight to the collided track (Fig. 12). The uncollidedtrack is put on the cell boundary (the point intersectedby the cell boundary and the track direction), and thecollidedtrack’scollisionsite is sampled in the usualwayexceptthat the collisionsite must now be sampled from.a conditional probability, the condition being that acollisionoccursat a distanceO< x <1.
A. Comments
1. Although the forced collision technique is nor-mally used to obtain collisions in optically thincells, it can also be used in opticallythick cells toget the uncollidedtransmission.
2. The weightcutoffgame is normally turned offin aforcedcollisioncell(seeMCNP Manual for excep-tional casesl).
3. The forcedcollisiontechnique decreasesthe historyvariance.but the time uer history increases.
4.5.
More than one collisi& can be ~orcedin a cell.f of Fig. 12is alwaysthe distance from the point atwhich-the track ii split into its collided and un-collidedparts to the boundary. In Fig. 12,the splitis done upon entrance to the cell,but the split canoccurat an interior point as well (splitsat interior
16
CELL
PR13GR PROBL
2 23 34 45 56 67 78 89 9
10 1011 1112 1213 1314 +415 1516 1617 1718 1819 1920 2021 2122 22
TOTAL
TRACKSENTERING
223711861325149114151398141814451464140514501492156515771611163216011484652
:
27852
POPULATION
218910971208135813101295131313201356130513441366145314651484151414891438652
40
25960
COLLISIONS COLLISIONS* WEIGHT
289828973166336832013302318833173522320831293362368336253696383133583236
00
~98;
/NUMBEROF COLLISIONSPROCESSED]THE SAME,BUTABOUTHALF THENUMBER OF PARTICLES RUN
\
)
(PERHISTORY)
2.1476E+O01. I189E+O06.4234E-013.0964E-GI1.3708E-017.8890E-023. IO16E-021.6468E-027.0522E-032.9794E-031.4564E-036.9754E-043.3628E-041.5533E-047.1971E-053.2251E-051.1371E-055.0066E-06o.0.0.
4.4947E+O0
APPARENTLY
NUMBERWEIGHTEO
ENERGY
1.8374E+O01.2427E+O08.5774E-019.1416E-017.9773E-015.9432E-017.7036E-016.4889E-018.0370E-016.6734E-015.5143E-017.1905E-016.4563E-015.9601E-015.9638E-016.1970E-017.3876E-017.1488E-011.7044E+O01.7145E+O0o.
FLUX AVERAGE AVERAGEWEIGHTED
ENERGY
5.6742E+O04.0314E+O03.1891E+O03.1710E+O02.9849E+O02.5009E+O02.7295E+O02.4403E+O02.5044E+O02.4093E+O02.1809E+O02.2514E+O02.1268E+O02.0087E+O02.0357E+O02.0445E+O02.2109E+O02.3301E+O03.4031E+O01.7156E+O0o.
TRACK WEIGHT(RELATIVE)
1.2206E+O01.3540E+O01.4630E+O01.3746E+O01.3433E+O01.4377E+O0t.2648E+O01.2894E+O01.1811E+O01.1671E+O01.2476E+O01.1605E+O0t.i339E+O01.1594E+O01. II15E+O01.0851E+O09.9903E-019.6091E-011.4500E+O07.7972E-01o.
TRACK MFP(CM)
6.7028E+O05.8642E+O05.5270E+O05.734iE+O05.6437E+O05.2856E+O05.5940E+O05.3027E+O05.4445E+O05.3589E+O05.1541E+O05.3834E+O05.1845E+O05. I098E+O05.1709E+O05.3119E+O05.5864E+O05.6517E+O01.0000+1235.8049E+02o-
THEWEIGHTCUTOFFANDIMPLICITCAPTUREDID—REDUCETHEHISTORY VARIANCE, BUT LOST SINCE THETIME PER HISTORY INCREASED TOO MUCH/
/
\/REDUCED RELATIVE ERRORFORSAMEPARTICLES \
TALLY FLUCTUATION CHARTS v\TALLY 1
J
\
TALLY 4 TALLY 5NPS MEAN ERROR FOM MEAN ERROR FOM MEAN ERROR FOM1000 4.45i47E-07 .2634 49 7.21958E-14 .9995 3 0. 0.0000 02000 5.474i8E-07 .2031 42 5.86343E-14 .7254 3 0. 0.0000 02099 5.61666E-07 .1933 37 5.58688E-14 .7254 2 0. 0.000Q o
/
************************************ ********* ***************************DUMP NO. 2 ON FILE RUNTPG NPS = 2099 CTM = .61
~AppEARswoRsETHAN FOM=50LASTTIME; STATISTICS STLLNOTVERY RELiABLE
NEXT TIME REMOVE WEIGHT CUTOFF AND IMPLICIT CAPTURE
Fig.11.Weightcutoff(.5.25)andimplicitcapture.
.
UNCOLLIDED WEIGHT we- al
COLLIDED WEIGHT
/
INCOMING WEIGHT W
Fig. 12. Forcedcollisionprocedure.
points normally occur when more than one col-lisionper enteringtrack is forced).
B. Caution
Becauseweight cutoffs are turned off in forced col-lision cells, the number of tracks can get exceedinglylargeif there are severaladjacent forcedcollisioncells.
C. ForcedCollisionsAppliedto the%mnpleProblem
Recallthat the point detectortally(tally5)wasnonex-istent becausethere were no collisionsin cell21. Figure13showsthe effectsof forcing’one collisionin cell21 inaddition to energy cutoffi refined geometry split-ting/Russian roulette, and energyroulette. Note that 44tracksentered cell21 and there were44 collisionsin cell21.Alsonote that the point detector tally is now obtain-ing contributions. Thus, the forced collision has reallyhelped the point detector tally. The trouble now is notthe lack of collisionsfrom tracks that enter cell 21, butrather the small number of particles that enter cell 21.Anglebiasingin some form isrequired to preferentiallyscatterparticlesinto cell21.
X. IDXTRAN
The DXTRAN technique and source angle biasingare currently the only angle-biasing techniques inMCNP. Unlike source angle biasing, DXTRAN biasesthe scatteringdirectionsas wellas the sourcedirection.
Before explaining the DXTRAN theory, I will firstloosely describe what occurs. A typical problem inwhich DXTRAN might be employed is much like thesampleproblem;a small region(for example,cell21) isbeing inadequately sampled because particles almostneverscattertoward the smallregion.To amelioratethissituation, the user can specify a DXTRAN sphere (inthe input file) that encloses the small region. Uponparticle collision (or exiting the source) outside thesphere, the DXTRAN technique creates a special“DXTRAN particle” and deterministically scatters ittoward the DXTRAN sphere and deterministicallytransports it, without collision, to the surface of theDXTRAN sphere (Fig. 14).The collisionitselfis other-wisetreated normally,producinga non-DXTRAN par-ticle that is sampled in the normal way, with no reduc-tion in weight.However, the non-DXTRAN particle iskilledif it tries to enter the DXTF4N sphere.
The subtletyabout DXTRAN is howthe extra weightcreated for the DXTRAN particles is balanced by the
18
InZ1-z
Ozo
.cli-In
ticnl-lu
ll+A
3Z
AO*IZ
vuLL
..
..
..
..
..
.,,..,.
.—
!“.—.
Wm
mm
mm
mm
mm
mm
mm
mm
mm
vm-
..
..
..,..,,
..
..—
..-.—
.---.
----yyoW
UJJ
N*m
W*O
IU
-J*U
)mmln
W
-..$)??
WW
UJ
mom
+0-J
L-9
-Wm
lt-
btcl
,..W
(DW
00.yy?U
.!(4JU
Jm
mN
Ulom
-rC
9-f
m.+.
..
-.~
-mm
?yoU
Jhltl!
r-ma
.(OC
OC
5m
mC
)(9O,..
.ei.
wm
**wlnltY
lf)w-
$)$)????0-
Ulu
uu
lulu
iilA
e4c4
u2m
e40w
wm
lnvomqo
FJ
Wu-lm
ciwull-
mo?y-~
mm
co.
..
..
..
+m
-r-m.m
emo
z%
u........
..
..
..
-1<i-
\aw
it-m-s-
If)II
z::ao~
>um
mJE
u)~
-1u,.
i--m
t-++
..
.L
ILA
.-mm
co.m
w.
fmln
-w
vm.,.
..
.
l-r-..1-JU
A*Om
m.r--sW
v..-m
IL
Iit-Nmwm
ctwm
mom
moc+
omm
-omw
mO
@J-fM
W-W
LO
cOm
m*m
i-C90m
um
owm
m-om
mcom
+t-h
r-mgejN
.-----ooooooooo.
..
..
..
..
..
..
.
tomacoIi~z\1-a<Q
**-1W●W
****II**=*l-*U**1*******l-*-*W*-*C
9**E
n**IA*O
..*Z*********~**L*Q*6*Z*3*U**L
U*A*H*IL:2*O*●
N***.*O*Z**Q*=*3*O
19
[)XTRAN
//
.. ~ ““- . .~ “ -— — — — —- —, — — —
COLLISION POINTPOINT SAMPLED
ON DXTRAN SPHERE
FPARTICLE D!RECTIO’N BEFORE COLLISION ( U,V,W )
\
1. A point on the DXTRAN sphere is sampled.
2. A particle is scattered towards the selected point.3. The particle’s weight is exponentially decreased by the optical path and
adjusted for bias in the scattering angle.4. The original particOe is sampled in the normal way (with no reduction in
weight).5. If the original particle tries to enter the DXTRAN sphere, it is
terminated.
20
Fig. 14. DXTRANconcept.
weight killed as non-DXTRAN particles cross theDXTRAN sphere. The non-DXTRAN particle is fol-lowed without any weight correction, so if theDXTRAN technique is to be unbiased, the extra weightput on the DXTRAN sphere by DXTRAN particlesmust somehow(on average)balance the weightof non-DXTRAN particleskilledon the sphere.
A. DXTRANViewpoint#1
One can view DXTRAN as a splittingprocess(muchlikethe forcedcollisiontechnique)whereineach particleis split upon departing a collision(or sourcepoint) intotwo distinctpieces:
1. the weight that does not enter the DXTRANsphereon the next flighteitherbecausethe particleis not pointed toward the DXTRAN sphere orbecause the particle collides before reaching theDXTRAN sphere,and
2. the weightthat enters the DXTRAN sphereon thenext flight.
Let WObe the weight of the particle before exiting thecollision,let PIbe the analogprobabilitythat the particledoes not enter the DXTRAN sphere on its next flight,and let pzbe the analogprobabilitythat the particledoesenter the DXTRAN sphere on its next flight.The par-ticle must undergo one of these mutually exclusiveevents, thus PI + p2 = 1. The expected weight notentering the DXTRAN sphere is WI= WOP1,and theexpectedweightentering the DXTRAN sphere is W2=WOp2.Think of DXTRAN as deterministicallysplittingthe originalparticlewith weightw. into two particles,anon-DXTRAN (particle 1) particle of weightW1and aDXTRAN (particle 2) particle of weight W2. Un-fortunately,thingsare not quite that simple.
Recallthat the non-DXTRAN particleis follwedwithunreducedweightWOrather than weightWI= w. pl. Thereason for this apparent discrepancy is that the non-DXTRAN particle (#1) plays a Russian roulette game.Particle 1’sweightis increasedfrom WIto WOby playinga Russian roulette game with survival probability pl =wl/wO.The reason for playing this Russian roulettegameis simplythat PI is not known, so assigningweightW1= pl WOto particle 1 is impossible. However, it ispossible to play the Russian roulette game withoutexplicitlyknowingPI. It is not magic,just slightlysubtle.
The Russian roulette game is played by samplingparticle 1 normally and keeping it only if it does notenter (on its next flight) the DXTRAN sphere; that is,particle 1survives (by definition of p,) with probabilityP1.Similarly,the Russian roulettegameis lost if particle1enters (on its next flight)the DXTRAN sphere;that is,particle 1 loses the roulette with probability p2.Now Irestate this idea. With probability PI, particle 1 has
weightWOand does not enter the DXTRAN sphereandwith probability p2, the particle enters the DXTRANsphere and is killed. Thus, the expected weight notenteringthe DXTRAN sphere is WOPI + O”P2= W1,asdesired.
So far, this discussionhas concentrated on the non-DXTRAN particleand ignoredexactlywhat happens tothe DXTRAN particle.The samplingof the DXTRANparticlewillbediscussedaftera secondviewpointon thenon-DXTRAN particle.
B. DXTRANViewpoint#2
If you have understood the first viewpoint, you neednot read this viewpoint. On the other hand, if the firstviewpointwasnot clear,perhapsthis secondone willbe.
This secondwayof viewingDXTRAN doesnot seeitas a splittingprocessbut as an accountingprocesswhereweightis both created and destroyed on the surface ofthe DXTRAN sphere.In this view,DXTRAN estimatesthe weightthat shouldgo to the DXTRAN sphereuponcollision and creates this weight on the sphere asDXTRAN particles.If the non-DXTRAN particle doesnot enter the sphere, its next flightwill proceed exactlyas it would have without DXTRAN, producing thesame tally contributions and so forth. However, if thenon-DXTRAN particle’s next flight attempts to enterthe sphere, the particle must be killedor there wouldbe(on average) twice as much weight crossing theDXTRAN sphereas there shouldbe, the weightcrossingthe sphere having already been accounted for by theDXTRAN particle.
C. TheDXTRANParticle
Although the DXTRAN particle does not confusepeople nearly as much as the non-DXTRAN particle,the DXTRAN particleis nonethelesssubtle.
The problem is how to sample the DXTRAN par-ticle’s location on the DXTRAN sphere. One cannotafford to calculatea cumulative distributionfunction toselectthe scatteringdirection (3indicated in Fig. 14.[Theazimuthal angle is sampled uniformly in (0,2rc)]. Thiswould essentially involve integrating the scatteringprobability density at each collision. Instead of sam-pling the true probability density, one samples anarbitrary densityand adjusts the weightappropriately.
As indicated above, a point on the DXTRAN spherecan be selectedfrom any density function because theweightof the DXTRAN particleis modified by
true densityto selectpoint p,densitysampled to selectp,
INNER CONEANGLE 01— COs 81 = qI
DXTRANOUTER CONE
— — — — — — — —.— —
inner DXTFIAN sphereCOLLISION POINT
PARTICLE DIRECTION BEFORE COLLISION ( U,V,W )
q= Cos)(0)
Fig. 15. SamplingtheDXTRANparticle.
This is easy to do because the true scattering densityfunction is immediately available even if its integral isnot. MCNP arbitrarily uses the two-step density de-scribed below. In fact, the inner DXTRAN sphere hasonlyto do with this arbitrary densityand is not essentialto the DXTRAN concept.
MCNP samples the inner cone uniformly in (TII,l),and the outer cone uniformly in (qO,ql)(Fig. 15).How-ever, the inner cone is sampled with five times theprobabilitydensity that the outer is sampled. That is tosay the inner cone is taken to be five times as importantas the outer cone. Further mathematical details aregiven in the MCNP manuall and will not be discussedhere.
After the scattering angle has been chosen, theDXTR4N particle is deterministically transported tothe DXTRAN spherewithout collisionand with weightattenuated by the exponentialof the opticalpath.
D. InsidetheDXTRANSphere
SOfar, only collisionsoutside the DXTRAN spherehave been discussed.At collisionsinside the DXTRANsphere,the DXTRAN gameis not played*becausefirst,the particle is already in the desired region and second,it is impossibleto definethe angularcone of Fig. 14.
E. Terminology-RealParticle,P’seudoparticle
InX-6 documentation,atleastthroughthe April 1981MCNP Manual,l the DXTRAN particle is called a
*IfthereareseveralDXTRANspheresandthecollisionoccurs,inspherei, thenDXTRANwillbeplayedforallspheresexceptspherei.
22
pseudoparticleand the non-DXTRAN particle is calledthe original or real particle. The terms “real particle”and “pseudoparticle” are potentially misleading. Bothparticles are equally real; both execute random walks,both carry nonzero weight, and both contribute totallies. The only stage at which the DXTRAN particleshould be considered “psuedo” or “not real” is duringcreation. A DXTRAN particle is created on theDXTRAN sphere, but creation involves determiningwhat weight the DXTRAN particle should have uponcreation. Part of this weight determination requirescalculating the optical path between the collision siteand the DXTRAN sphere. MCNP determines the op-tical path by tracking a pseudoparticle from the col-lision site to the DXTRAN sphere.This pseudoparticleis deterministically tracked to the DXTRAN spheresimply to determine the optical path; no distance tocollisionis sampled,no talliesare made, and no recordsof the pseudoparticle’spassage are kept (for example,tracks entering). In contrast, once the DXTRAN par-ticle is created at the sphere’ssurtlace,the particle is nolonger a pseudoparticle; the particle has real weight,executesrandom walks,and contributes to tallies.
F. Comments
1.2.
3.
DXTRAN sphereshave their own weightcutoffs.The DD card (by default) stops extremely low-weighted tracks by roulette. See the manuall forhow this is accomplished.Strongly consider producing DXTRAN particlesonly on some fraction of the number of collisions,as allowedby the DXCPN card.
G. CAVEATS
1.
2.
3.
4.
DXTRAN should be used carefilly in opticallythick problems. Do not rely on DXTRAN to dopenetration.If the source is user-supplied, some provision(SRCDX, page 263 of the MCNP manual’) mustbe made for obtaining the source contribution toparticleson the DXTRAN sphere.Extreme care must be taken when more than oneDXTRAN sphere is in a problem. Cross-talk be-tween spherescan result in extremely low weightsand an explosionin particle tracks.A different set of weirzhtcutoffs is used inside theDXTRAN sphere. -
H. DXTRAN Appliedto the Sample Problem
Recall that there was a problem gettingenough parti-cles to scatter in the direction of cell 21. To solve this
problem, a DXTRAN sphere was specifiedjust largeenough to surround cell 21 (Fig. 16). If a largerDXT’RANsphere were used, some DXTRAN particleswould miss cell 21 and this would be less eflicient. If asmaller DXTRAN sphere were used, it would bepossiblefor a non-DXTRAN particle to enter cell 21,resulting in an undesirable large weight fluctuation incell 21. Note also that the inner and outer DXTRANspheres are coincident. This choice was made becausespecifyingdifferent spheres would introduce a five-to-one weightvariation even though all particles enteringcell21 are about equallyimportant.
I. Discussion
Note from Fig. 17 that DXTRAN did have the de-sired effect; the tracks entering cell 21 have increaseddramatically and the FOMS for tallies 4 and 5 haveincreased by a factor of 7. However, note that theparticles-per-minutenumber has decreased by a factorof 4; this is reflectedin a factor of 4 decreasein tally 1’sFOM. It would be wonderful if DXTRAN did not slowthe problem down so much. Fortunately in some cases,’a little thinking and judicious use (describedbelow) ofthe DXCPN card can alleviatethis speed problem.
Recall the caveat about using DXTRAN carefullyinoptically thick problems, in particular, not to rely onDXTRAN to do the penetration. Geometry splittinghasdonewellat penetration, so DXTRAN is needed mostlyfor the angle bias, as is desirable. However, at everycollision,regardlessof how many mean free paths the
Fig. 16. DXTRANsphere.Theinnerand outerspheresareidenticalbecausespeciijhgdifferentsphereswouldjust createweightfluctua-tion.
23
collision is from cell 21, a DXTRAN particle isproduced. DXTRAN particlesthat are many free pathsfrom the DXTRAN sphere will have their weightsex-ponentially decreased by the optical path so that theirweightsare negligibleby the time they are put on theDXTRAN sphere.MCNP automatically(unlessturnedoff on the DD card) plays Russian roulette on theDXTRAN particlesas their weight falls exponentially,becauseof transport, belowsomefractionof the averageweight (on the DXTRAN sphere). This provides theuser some protection against spending a lot of timefollowing DXTRAN particles of inconsequentialweight. However, there is a better solution for thesampleproblem.
Although the DD card will play roulette onDXTRAN particles as they. are transported throughmedia to the DXTRAN sphere, it still takes time toproduce and followthe DXTRAN particles until theycan be rouletted. It is much better not to produce somany DXTRAN particles in the first place. MCNPallowsthe user (on the DXCPN card) to specify,by cell,what fraction of the collisionswill result in DXTRANparticles.Everythingis treated the same exceptthat if pis the probabilityof creatinga DXTRAN particle, thenwhen a DXTRAN particle is created, its weight ismultipliedby p–l, thus making the game unbiased.Thedestruction game is unaffected; regardless of whetherthe sampling produced a DXTRAN particle, the non-DXTRAN particle is killed if it tries to enter theDXTRAN sphere.
As usual, this new capability requires even moreinput parameters; that is, the entries on the DXCPNcard. Beforedespairingunduly, note that the entries onthe DXCPN card are not highlycritical,and the userhasalready gained a lot of useful information in thegeometry-splittingoptimization.
Table I showsthe DXCPN probabilitiesthat I chosefor the sample problem. Note three things from thistable.
1.
2.
3.
24
Near the top of the concretecylinder(cells18and19) every collision creates a DXTRAN particle(p= 1).As the cellsget progressivelyfarther (cells 12-17)from the DXTRAN sphere, p gets progressivelysmaller by roughly a factor of 2, chosen becausethe importance from cell to cell decreasesby fac-tors of about 2.Not much thought was spent selectingp’s for cells2-11 because these cells contribute almost noweight to the DXTRAN sphere. Thus withinreason, almost any values can be selected if theyare small enough that not much time is spentfollowingDXTRAN particles in cells 2-11. Notethat even p = 0.001willnot totally precludecreat-ing DXTRAN particles because there are
2000-3000collisionsk each ofcells2-5,wherep =0.001.
Beforeexaminingwhat happened when the DXCPNcard was used, I would like to digressand use item 3aboveas a specificexampleof a generalprinciple.Whenbiasing against random walks of presumed. low im-portance, alwaysmake sure that at least a few of theserandom walksare folIowedso that if the presumption iswrong, the statistics will so indicate by bouncingaround. As an example, I fully believe that p = 10-6wouldbe appropriatein cell2, but I chosep = 10-3.HadI chosen p = 10-6, probably no DXTRAN particleswould be produced from collisions in cell 2. Thus iftlese DXTRAN particles turn out to be a lot moreimportant than anticipated, the tally may be missingasubstantial contribution with no statistical indicationthat somethingis amiss. By choosingp = 0.001 in cells2-5, I cause the MCNP to produce approximately tenDXTRAN particles by the 10,000 or so collisions incells2-5 (seeFig. 17).Following10DXTRAN particlesis a very small time price to pay to be sure that they arenot important. If the problem were to be run longenoughthat therewouldbe 107collisionsin cell2, then Iwould not hesitate to use p = 10–6because someDXTRAN particleswouldbe produced.
J. ResultsofUsingDXTRANwiththe DXCPNcard
The resultofaddingthe DXCPN card is shownin Fig.18.Note that all FOMSimproved by better than a factorof 2. The histories per minute increased from 1560to4395when the DXCPN card wasadded, but 4395is stillslowerthan the 6858without DXTRAN. The FOM fortally 1, although almost three times as good as thatwithout the DXCPN card, is nonethelessstill less thanthe no DXTRAN FOM of 45.This is an exampleof thegeneralrule:
Increasingsamplingin one re@onin generalis at theexpenseof another region.
In the sampleproblem,we have decided to increasethesamplingof cell21 at the expenseof cells2-19.Overall,however, DXTRAN has clearly improved the calcula-tion.
XI. TALLY CHOICE, POINT DETECTOR VER-SUS RINGDETECTOR
Recall from the introductory section on variancereductionthat the FOM is affectedby the tally choiceaswell as by the random walk sampling. So far, I havetinkered only with the random walk sampling; now,supposeI tinker with the tally.
Considertally5,the point detectortally.Note that thesampleproblemis symmetricabout the y-axis,so a ring
000000
OO
OO
OO
OO
CO
CY
CW
OI
oooooo880000000000e40m+++++++++
+++-I--F+*++W
+-
Uw
wu
ww
ww
uu
ww
ww
ww
ww
+w
+m
mm
wm
wcn
pm
wm
t-mw
wm
m.om
m-m
wm
wm
=tow
mm
mw
mm
maow
8m
mw
o-wow
mm
m~w
-mm
mm
mw
~t.omm
mvm
~m~m
-v~~8%
%...,
,.,.,,.,,.
.,.,,,,
Wm
mm
wm
mm
mm
mqqm
mm
mm
-br
U-I
“>”Z
1-ci010.cJ
l-(n
I-llnU
W1-1
J>x
-1O*E
vun.
u-l
z0l-(u
-l!+-1dc1uzc11+
1-
53n1?
00---Ne4oJm
mcom
ww
mm
mw
000000000000000000++1111111
Illlll,a,Lu
ulu
uu
wu
uu
uu
uu
uu
uu
wm
mcm
wcooot-m
mw
wm
.moom
-t.wm
wm
o-ut~
-omm
mm
w~
mw
mm
mm
wm
~m
ovmm
my~
~ow
omom
mt-m
mm
owm
ay~
a
wcot-
/
In
C9m
cl-..
:-*:*II
**S*i-*V.*********+*-*@
l*IX*C
4*:**
II*
I
*VI
*nO
lw●
z-.
\/
*****m
m:
%2
*C
4tl
,.z.io
*Q*l-*Z*3*K**W*.J*H*IL*:5*
zaat-
G
25
t+m
CELL
PROGR PROBL
2 23
: 45 56 67 78 89
1:;: 1112 1213 1314 1415 1516 1617 1718 1819 1920 2021 2122 22
TOTAL
TRACKSENTERING
11938529250104827441441774114411441054112429343844337431243654274424838674927
152231283
107616
POPULATION COLLISIONS COLLISIONS NUMBER* WEIGHT WEIGHTEO
(PER HISTORY) ENERGY
11744491846514469407338163780380338333803394840404oi739774059398239353749
1822430451
1283
130555
1379611952li384108479805950795009158929792029827975996219935
10039986299468538
1559:o
197573
1.9875E+O01.0727E+O05.7591E-012.7700E-Oi1.2674E-016.2377E-023.2883E-021.5241E-027.4165E-033.2429E-031.7202E-038.0545E-043.6893E-04i.7810E-047.9894E-053.6959E-051.6748E-056.2137E-06o.6.5663E-11o.
4.1643E+O0
1.9553E+O01.1897E+O08.6965E-017.9325E-017.8981E-017.9364E-Of7.5226E-017.2149E-017.0749E-017.5649,E-016.9620E-016.5721E-016.6088E-016.6189E-017.0947E-016.3603E-016.6484E-017.0403E-011.2396E+O01.7785E+O07.0419E-01
FLUXWEIGHTEO
ENERGY
5.8909E+O04.0414E+O03.3229E+O03.0387E+O02.9908E+O02.8062E+O02.5813E+O02.5273E+O02.4918E+O02.4433E+O02.3424E+O02.2911E+O02.2587E+O02.1957E+O02.1324E+O02.0865E+O02.0783E+O02.2236E+O03.2943E+O04.1554E+O02.7866E+O0
AVERAGETRACK WEIGHT
(RELATIVEI
1.3213E+O01.6629E+O01.9476E+O02.0971E+O02. i426E+O02.2181E+O02.3164E+O02.3520E+O02.3750E+O02.3664E+O02.4325E+O02.4785E+O02.4924E+O02.5004E+O02.5102E+O02.5629E+O02.5527E+O02.4620E+O07.9810E-019. I074E-043.6840E-05
AVERAGETRACK MFP
(CM)
6.8365E+O05.9855E+O05.6454E+O05.5575E+O05.6477E+O05.5340E+O05.4448E+O05.4441E+O05.4572E+O05.4819EiO05.3524E+O05.3043E+O05.3570E+O05.307iE+O05.3140E+O05.2996E+O05.3146E+O05.45iiE+O01.0000+1237.3805E+021.0000+123
SUBSTANTIALLYIMPROVED
TALLY 1NPS MEAN ERROR FOM1000 ~.2j356E-07 .3667 2720003000400050006006700080009000
1000011000li427
6.41539E-075.70257E-076.76150E-076.76891E-076.73265E-076.89040E-076.86656E-077.04305E-077.25099E-077.00085E-077.32339E-07
.2949
.2393
.1863
.1679
.1516
.13781262
:11671093
;1046.1049
26273232323435363636
/34
TALLY 4MEAN
1.50695E-131.06818E-131.02834E-131. I1525E-131.13338E-131.14934E-131.14298E-131.17711E-131. IBI17E-131.22116E-131.18453E-131.22412E-13
ERROR.325!.2461
1982:1622.1465.1312.1198.1095
::%.0948.0946
7 TALLY 5FOM MEAN
34 i.06148E-i66.85473E-17
% 6.49562E-i743 6.79i32E-f742 6.89970E-17’43 6.74936E-1745 6.74760E-1746 6.8725iE-i747 6.97952E-~744 7.09365E-i745 6.88873E-1742 7.21438E-1715LAs-rTN4E
ERROR.3673.2928.2387.1910.1686.1510.1359.1245.1161.1114.1060.1049
t’FOM
2727273131333436363536
**********************************fi**************************************DUMP MO. 2 ON FILE RUNTPH
/
NPS = 11427 CTM = 2.60 PARTJMIN= 4395NO DXTRAN =6858
LMPROVED OVE!?DXTRAN w/0D~CP~=12 DXTRAN~IeDxCPN =156QLESS THANNODXTRAN=45~INcREAslNG sAMPL/NG INONIE FtEGtONGENERALLy
ISATTHEEXPENSEOF ANOTHER REGtONm
Fig.18. DXTRANwithDXCPFlcard.
000000000
000000000
++-4
-+-+
+++.+
Ulu
.lluu
lulu
luu
(u-v~
cNFN
IDm
+O
)mo.m
wqln
w.t.o.m
coNw
mLn
maw
mm
vvtw
Oooooom
wm
ooooooo/oN
●1-+
+-+
++V
.+P
UIK
lullu
ldlu
+u
+ln
mcol.m
r-ooo
:gj~:~
~:g
..
,.
.,
,.
,.
,,
..
..
,.–,
.—,
Wm
mm
mm
mm
mm
mfflm
mm
mm
m.+
-
000000000
000000000
++++-+
++++
UJLU
UJLU
UIU
JU
JU
JLU
Qlcn
-cmm
ou)m
oK
JV
t-omoin
ww
CJ1-t.col-m
cdu
lmm
wm
o.Nm
mm
..,,.
..
..
.rs-.@J
**~~~
Oooooo-=rm
000000000
++-+
+++111
Ulu
.lulu
,ll,uu
lu,lu
lwcL
,olslulw
Nm
c.4fn
wN
~w
-tlnm
sroC
Tw
sru)vow
mal
Vm
wln
u)w
mm
ul
..
..
..
..
..
..
.
000
yyy
Uw
wd’w
.C
x-.g
.00.
$?$U
wu
W(-O
F-m
mO
lnw
(n-m
---???W
uu
l-ohN
mw
W*N
Ww
m
l-&bh
**t-l---.-.s-..!-.
In1111111,
t-r-wt-
----111!
UIL
IUJU
,Iom
~m
c9m
N.
mw
cowU
Iww
t.r-cob
t-
W“(iw
”li
aoco-Nt-.+
mm
qu)o
Om
bw
..w-ow
mw
mu
omm
ww
mm
.omm
mam
t-4-..--.-ooo
U1
...
..
..
..
..
.
mm
comr.-.1
111
WW
IJJW
:CO
:C
-J
*WO
.F
aFw
----,..,
----
Xf-a
=rb
mm
mm
mom
m+ommmmmmmc.ymwmm
ILctvmw
r.mt-m
wm
mw
owu
-Jw
oNcN
otJIc.J
*@Y
8tx-m
ot.mqm
--ooofilv:.x
---------,...
..
..
..
**O*W!$4●*
II
ii;In*a
.*z**********x*u*Q
.*+*z*>*a**U
J*J*U*L
L**z*o*S
W***.*o*z**L*E*3*n
27
TABLE I. DXCPNCardEntries
Cell Probability
23456789
101112131415161718192021
22
0.0010.0010.0010.0010.010.010.010.010.010.010.0150.020.040.080.20.411VOID, no collisionsINSIDE SPHERE, no DXTRAN
gamepIayedVOID. no collisions
detectorcan be used instead ofa point detector.The ringdetectorestimatesthe averageflux on a ring rather thanthe flux at a point, but because the sample problem issymmetric, these tallies (on average)will be the same.The ring detector gives lower variance estimates thanthe point detector, especiallyiflunlike the sampleprob-lem, the detectors are embedded in a scattering me-dium. On average,collisionsare closerto a ring detectorthan to a point detector, so the ring detector bettersamples the close collisions that tend to trounce thepoint detector statistics.Particularlyimportant in someproblems,but not this sample problem, is that the ringdetector has finite variance even in a scattering me-dium. The point detectordoes not.
For the sampleproblem, I chose a ring of radius 200cm about the y-axis such that the ring detector wentthrough the point where the point detector had been.The resultsare shownin Fig. 19.Note that everythingisabout the same as with the point detector, except thatthe ring detector’sFOM has increasedfrom 34 (Fig. 18)to 41.More differencewouldbe seenif the detectorwerein, or closeto, cell21.
XII. BIASINGTHE SOURCE
No attempt hasbeen made to bias the sourcealthough1. sourceparticlesmovingdownward (–$7)are unim-
portant becausethey immediatelyescape,and
2. high-energysource particles (14 MeV) penetratebetter than low-energysourceparticles(2 MeV)soare more important.
MCNPhas two typesof sourcedirectionbias that willbeemployecl,followedby sourceenergybias.
A. ConeBias
Cone biasing,a type of angular biasing, is illustratedin Fig. 20. A cone is specifiedthat divides the angulardomain into two pieces,one inside and one outside thecone.The user then specifiesthe fraction of particles tobe started inside the cone and outside the cone. Allparticles started inside are of one weight; all particlesstailed outside are of another (in the absence of othersourcebiasing,for example,sourceenergybiasing).Oneconsequenceof all particles inside the cone having oneweight and all particles outside the cone having a dif-ferent one is that there is weight discontinuity at thecone surface.This weightdiscontinuity should be con-sidered before using heavy cone biasing. Exponentialsourcebiasing,discussed in Sec. XII-B, should be con-sideredif the conebias weightdiscontinuityis too large.
Figure 21 shows the effectsof using cone biasing tosend 99Y0of the particlesin the +? half-space.Note thatthe FOMSare roughlythe same as before cone biasing.Indeed, the only major differenceis that the number ofparticlesstartedhas droppedby a factorof 2, as mightbeexpectedbecausealmost all of the time is spent follow-ing particlesmoving in the +j direction. No improve-ment occurredbecause the source sampling is very fastand it doesnot take longfor sourceparticlesgoingin —~direction to die. Stated another way, both runs hadabout 6000 particles sampled in the +$ direction, soboth runs gave roughlythe same results. The cone biassaved only a small amount of time ntit sampling the6000particlesthat would have gonein the –~ direction.
‘B.ExponentialSourceBiasing
In addition to the conebiasjust discussed,MCNP hasa continuous anglebias called exponential source bias-ingbecausethe sampled density is an exponentialin thecosineof the anglewith respect to a specifiedreferencedirection. That is, the probability density function forexponentialsourcebiasingis
p(w)= Cekv (j.L= cose),
where k = user-selectedbiasing parameter 0.01 < ks 3.5and C = normalization constant C = k/(e~e-~).Table 11shows how the particle weightat some anglesvaries with k. Note that although the exponential anglebiasinghas no weightdiscontinuities,largeweightfluc-tuations can be introduced by setting k too large. For
28
REFERENCEDIRECTION
SPECIFY:
1. v = cos (@) FOR FAVORED CONE
2. FRACTION OF PARTICLES STARTED INSIDE CONE
COMMENTS:
1. ALL PARTICLES
2. ALL PARTICLES
example,with k = 3.5, the weight ratioand 0 = 180°is 1094.
INSIDE CONE HAVE IDENTICAL WEIGHTS
OUTSIDE CONE HAVE IDENTICAL WEIGHTS
Fig.20. ConeBias.
between (3= 0°
I chosethe exponentialbiasingparameter k= 2 on thebasis of Table II. Recall that any particle departing thesource in the –~ direction (e > 90°) will be killedimmediately. Thus I confined my attention to theweightvariation between 6 = 0°and 0 = 90°.Fork= 2,there is a factorof about 8 fluctuation in weightbetween(i)= O“and 90°. Experience indicated that a sourceparticle at e = 0° might be eight times as imortant as asource particle at 90°. Maybe 8 was not a particularlygoodguess,but I wouldbe highlysurprisedif the “right”ratio werenot within a factor of 3.
Figure 22 shows the effects of exponential sourcebiasing.The FOM columns indicate no drastic changeand probably a small degradation in calculational effi-ciency.Thus source angle biasing did not appear effec-tive for the sample problem. However, a conferenceparticipant (John Hendricks) suggested that sourceanglebiasingmight have worked better with the weightwindow technique (Sec. XIII) than with the geometrysplitting/Russian roulette technique used here. I shallhave more to say about Hendricks’ suggestionin Sec.XIV.
C. SourceAlterationinthe SampleProblem
The runs so far have been with an isotropic sourcewith the followingenergydistribution:
1. 25%of the particlesstarted at 2 MeV.2. 25Y0of the particlesuniformlydistnbutedbetween
2 MeVand 14MeV.3. 50%of the particlesat 14MeV.
In preparing this report I had intended to use 50’Yoat 2MeVand 50%at 14MeV,so sourceenergybiasingcouldbe tried on a simple case. After discovering the inputerror that arose from using the first energydistributionaboverather than the second,I decided that if the sourcewas goingto changeanyway,a more interesting sourcecould be used instead of the second distribution. Allsubsequentruns have 95Y0at 2 MeV and 5%at 14MeV,making it easy to demonstrate biasing in energy. Notethat this spectrum is much softer than the one usedbefore, so tallies will drop and the calculation willthereforebe more difficultthan before.
The first run with the new sourceusesall the success-ful variance reduction techniques (with identicalparameters) used for the sample problem with the oldsourceexcept energyroulette. Specifically,the first runwith the new sourceuses
29
CELL
PROGR PROBL
2 234 :5 56 67 78 89 9
to 1011 1112 1213 1314 1415 1516 i617 1718 1819 1920 2021 2122 22
TOTAL
TRACKSENTERING
6605564053305074468i44644115416143024424449846834671461447214653435939164486
13367$063
103827
POPULATION COLLISIONS COLLISIONS* WEIGHT
(PER HISTORY)
640652204934471843064090379938583985410541574329434042714384430940603823
1613426736
1063
123027
146951281212290112821014310110
926793229585
1023110257i022910751104941080211002
99528868
013672
0
205764
1.9658E+O01. I148E+O06.0646E-012.6844E-011.2446E-016.6575E-023.0420E-021.4746E-027.4080E-033.5739E-031.7292E-038.0300E-043.8336E-041.7918E-048.6520E-05
“3.9905E-051.6436E-056.4892E-06o.6.7894E-11o.
4.2059E+O0
NUML3kKWEIGHTEO
ENERGY
2.0739E+O01.1286E+O08.2161E-017.9748E-o~
8.0250E-017.3839E-017.6131E-Ot7.1952E-017.0524E-016.7007E-016.8539E-016.7528E-Oi6.9439E-016.6366E-016.0949E-016.1264E-Oi6.4482E-016.8815E-011.4905E+O01.5740E+O07.2051E-01
FLUXWEIGHTEO
ENERGY
6.0195E+O03.9936E+O03.2831E+O03.t278E+O02.9926E+O02.7183E+O02.6942E+O02.5393E+O02.4389E+O02.3164E+O02.2800E+O02.2984E+O02.2499E+O02. 1724E+O02.0604E+O0i.9942E+O02.0619E+O02. 1795E+O03.5288E+O03.9432E+O01.9923E+O0
AVtKAGtTRACK WEIGHT
(RELATIVE)
6.6003E-018.5805E-011.0041E+O01.0501E+O01.0843E+O01.1594E+O01.1432E+O01.1895E+O01.2186E+O01.2352E+O01.24t-iE+O0t.2314E+O01.2361E+O01.2611E+O01.3078E+O0i.3234E+O01.2907E+O01.2469E+O03.9695E-015.5103E-041.8421E-05
AVtKAtitTRACK MFP
(CM)
6.8857E+O05.9323E+O05.5831E+O05.6114E+O05.6203E+O05.4194E+O05.5398E+O05.4467E+O05.4138E+O05.3339E+O05.3575E+O05.4287E+O05.4183E+O05.3735E+O05.2351E+O05.2048E+O05.3094E+O05.4875E+O01.0000+1237.2941E+021.0000+123
ROUGHLYTH”E”SAME
ITALLY i + TALLY 4 TALLY 5
t
NPS MEAN ERROR FOM MEAN ERROR FOM MEAN ERROR FOM1000 9.62664E-07 .2587 28 1.52991E-13 .2256 37 8.70445E-17 .2290 362000 8.86500E-07 .1777 33 i.39i45E-13 .i59i 41 8.47413E-17 .i652 383000 7.21686E-07 .1543 32 1.18189E-13 1358 41 7.33522E-i7 .1420 384000 6.89222E-07 .1295 34 1.19786E-13 :1124 45 7.30212E-17 .1185 415000 6.49018E-07 .1225 31 1.15679E-13 1031 44 7.00919E-17 1077 406000 6.87974E-07 .1084 32 1.23255E-13 :0924 45 7.42021E-17 :0972 406049 6.82401E-07 .1084 1.22481E-13 .0922 45 7.37452E-17 .0970 40
**************************************************************************our4P NO. 2 ON FILE RUNTPF
>
NPS = 049 CTM = 2.60
- AS EXPECTED, ABOUT HALF
CONCLUSION: NO IMROVEMENT BECAUSE DID NOT TAKE LONG FOR SOURCE
PART!CLES GOING !N -$ D!!?ECT!ON ?O!3!E.
Fig. 21. Conebiasing-99%in+fhalf-space.
1.2.
3.4.5.
energycutoff,geometry splitting/Russian roulette (refinedparameters),forcedcollisionin cell21,DXTRAN with DXCPN probabilities,andring detector.
Figure23 showsthe resultsof the first run. Note that, asexpected,the talliesand FOMShave decreasedsubstan-tially. The geometry splitting could probably be im-proved somewhatto keep the “tracks entering” roughlyconstant.
TABLE II. Exr)onentialBiasin~Parameter
Cumulativek Probability Theta Weight
.01 00.250.500.751.00
1 00.250.500.751.00
2 00.250.500.751.00
3.5 00.250.500.751.00
06090
120180
0426493
180
0314870
180
0233753
180
0.9900.9951.0001.0051.010
0.4320.5520.7621.2303.195
0.245”0.3250.4820.931
13.40
0.1430.1900.2850.569
156.5
‘k= 2 waschosenbecausethe weightis approx-imately2 at 90”,whichis eighttimestheweightatOO;thisdoesnotseemunreasonable.
Beforeworryingabout optimizingthe geometrysplit-ting, I shall discuss the effect of source energy biasingbecause first, geometry splitting optimization has al-ready been illustrated, and second, the source energybiasingwill increase the energy spectrum of the tracks,making the average track penetrate better. Tracks withlonger free paths will need less splitting to keep thetracks entering approximately constant. In short, the“tracks entering” column in Fig. 23 can be expected toimprovebecauseof sourceenergybiasing.
D. SourceEnergyBias
MCNP allows biasing the source in the energy do-main aswellas in the angulardomain. In biasing,the SBcard is used with the S1 and SP cards. The S1 cardsupplies energy ranges, the SP card supplies analogprobabilities, and the SB card supplies the actualprobabilitiesused to sample the energy ranges. Beforeattempting a long run, look at the sourcebias informa-tion in the MCNP “outputand check that the weightmultiplier is not unreasonable.Figure 24 is an exampleof the source bias information from the run describednext.
Recallthat the natural sourceis 95%at 2 MeVand 5%at 14MeV. It is a goodguess(based on experience)thatthe 14-MeVsource neutrons are much more importantthan the 2-MeVsourceneutrons; therefore, I biased thesource to get 10%at 2 MeV and 90%at 14 MeV. The“weight multiplier” column in Fig. 24 shows that theratio of weightsis 171;that is, the sourceenergybiasingassumesthat 14-MeVneutrons are 171times as impor-tant as 2-MeVneutrons. This seems too high until oneconsidersthat 180 cm of concrete must be penetrated.The 14-MeVneutronscan probablypenetrate 171timesbetter. In any case,thousandsof neutronsare run, whichmeans that there will be hundreds of 2-MeV sourceneutrons.Thus the statisticscan indicatewhether 171ismuch too largebecause2-MeVsource neutrons are notprecludedby the sourcebiasing.
Figure 25 shows the results of the source energybiasing. All FOMS increased by a factor of 4 and, aspredicted, the “tracks entering” column has improvedsubstantially.Source energy biasing has definitely im-proved things, but could the same improvement beobtained using the energysplittingand roulette schemethat wassuccessfulearlier?
E. EnergyRoulette(WithoutSourceEnergyBias) Ap-pliedtothe SampleProblem
Figure 26 shows the results of removing the sourceenergybias and insertingthe energyroulette game:
50%survival crossing 5-MeV, l-MeV, 0.3-MeV,O.l-MeV and 0.03-MeVenergybounds.
The FOMSarea factor of2 better than the referencecase(Fig.23)that had no biasingin the energydomain, but afactor of 2 worse than the source energy biasing. The“tracks entering” column is flat deep into the concrete
cylinderbut decreasingvery fast at the sourceend. Thisdecreaseis probably becausethe 2-MeVparticles fail tosurvive the energy roulette game. Indeed, a look at thecreation and loss ledger (Fig. 27) tends to confirm thatenergy roulette is killing a lot of tracks. The energysplitting and Russian roulette are the “ENERGY IM-PORT” entries.
31
u41
1-
Oooooooooooooooooom
mm
OO
OO
OO
OO
OO
OO
OO
OO
OO
C.J
ON
+++-!-+
.++++
+++++++++-.+
-U
lwu
lulw
ulu
u,lu
!uu
luu
ldu
luw
u.1
+u
+.
o~~~a~-m
mm
mm
om~w
omom
o-o-m
mm
mw
=rm
~m
’wl-u
lmm
-owt.m
wlvls)cgg)lnql-tm
mqm
.lgqmo-
8om
mvm
*mN
mm
mN
wm
mm
*Dom
o.
..
..
..
..
..
..
..
..
,,..w
mm
mm
mm
mm
mm
mm
mm
mm
m.~
.
000
000
8++++
WLU
WLU
mtm
mw
NU
3F
0om
mci
LnFF
w
0000
yyyy
Lu
ullu
ul
Nt.
c91-
mm
cwr-
WC
Q-N
=rC
omcl
y?$W
LU
Wm
c4m
Coc-u
la)cn
a)N
om,..
.,..
-.-.-.,-
$)???
U.lw
uld
Cvin
wo
Ulc+
l-mw
a3e4
mm
mb-
..
..
..-.eoo-$l???:y?
Uu
luu
uw
ww
-~+t-w
mm
.Nm
mN
ov+
r-cv-mm
mm
m-m
wr.m
..
..
..
.
32
VI
Z1-ii
C!zo
uc3
l-In
I-41n
null-(-1
3Z
oo~yy?
141.u
ul
Om
cdm
omw
wm
mom
.,.-.-~
U)m
mw
???0U
UU
JJ
-30m
.l-w
Nm
Wafflln
m+w
a.
..
.W
C9.
U)
zol-l1-
<A3n0n
u
’45
vuauEl-
I-zu
-1-1UI
0
en0z
Et-w
O.in
wow
ixNs-
W..m
m.-
*11
z::aom
>W
qln_izm
r-2
0-a
,,1-
..e-
NU
)*-emm
om.s-0
0..
wm
m---
*********W+0
*W*D***
txmm
m-q
mobmom=tb
cxmmwm.o
UN---Y.
u......
**********U*n.
*l-*Z*2*IY**W*J*M*U
.**Z*O**N:*.*O*ZIL
L*E1:
000000000
000000000
++.+
++’+
+++
Uu
lLlu
Ju
!l.uLu
lu!u
*o=rm
oP3u
3N
ul
Ou
l-eln.ol-b
comcQ
FcJ
u3c9
w-
O-cu
mm
omw
o..,,
,,...m
=%
mm
mw
=t=
rm
000000000
000000000
+-h
+++++.+
+IU
LU
IJ.IL
UU
JU
!UJU
IUU
-mlm
t-ocJo.m
Cn
omu
locnm
cnu
lo-m
lmm
-coN@
l03*U
CO
C3U
CO
CO
U3
.,.,.Iim
’u+tAmLom
mm ‘m
@Jm
‘mom
;~y
Om
o000
Ou
-loO
wo
,..-*-
0000000000000000...W
Q3
000000000000000000000+-+
-!-++++++
+-+
+++++,,,
,,U
uu
ww
uu
ww
ww
uw
uu
mm
uu
uti
oooooooooooooooocnoo>@
Ioooooooooooooooom
Fu
.NO
ooooooooooooooocnt.u
-lmm
Oooooooooooooooom
ln.w
+,..,
..
..
..
..
..
..
..
!....
..
..
..
..
..
..
..
..
mw
mm
mm
mem
ww
ww
ww
ww
w~v~
m“>’
zl-lY0s0
uc5
l-W
)bfln
HU
H-J
3Z
O*LY
Clw
n.
VI
z0ul
.-!-10vz0.+a-1i?0c!.u
::VII
s;l-zw
I
mm
mm
m-..,-,-
LIY
JJLL
JA
AZmbor.
r.
awo
m-o
>uw)mmmm
Azmomw.
-1Fwmmln
<..,
,.
1-mmw.xm
mm
comm
mm
mm
mm
m.-.,-.,----------
UJlu
wtiw
z-or-t-b<
--LO
CJPW
>u
mqm
mtn
Azm
ot-mm
At-m
c-dcD-
a..,!
.
1-
w!D
wm
*
mm
mm
mm
mm
mm
mm
mm
mm
my??y)?ooooooooooo
1,1
,,8,,
,,,U
wlulutuw
wulw
lulaww
uww
wZ
ww
-t.wo*m
oow.-m
mw
oqa~w
wtow
mm
t--~w
wo.-m
m.
>m
c+t-c.im
wN
cn.mw
m.m
*mm
cja~om
t.or.~t.mqm
mw
wcnm
mm
-1N
mxroom
mw
mcw
mcom
vwm
qa
..
..
..
..
..
..
..
..
.
Oooooooooooooooom
moooooooooooooooo~
CL
oooooooooooooooooZc4
=r.w
m0N
vt9w
ocww
u2m
oe4m
.-...-.-@J~~~~m
m~
33
SOURCE = 1
w.P
SOURCE COEFFICIENTS
o.; 1.0000E-063 0.4 2.0000E+OO5 I.0000E+OO
S1 card SP card
/
BIASED SOURCE DISTRIBUTION O1
SOURCE SOURCE CUMULATIVE BIASEDENTRY VALUE
PROBABILITYPROBABILITY CUMULATIVE DENSITY
1/
2.0000OE+OO 9.500000E-01 I.0000OOE-01 9.500000E-012 1.40000E+OI I.0000OOE+OO I.0000OOE+OO 5.0000OOE-02
SB-card
1BIA’SED WEIGHT
PROBABILITY MULTIPLIER
I.0000OOE-01 9.500000E+O09.0000OOE-01 5.555556E-02
AVERAGE VALUE USING BIN MIDPOINTS = 2.6000E+O0
Fig.24. S~SP,aadSBcardsinsourceenergybias.
nu0)~mI
In“>”
ZI-IY
010
-+C
9+
UIH
mI-+
W.
-J3S
O*Evu
c1.
I/lz0.V
IHd-10uz0
000000000
000000000
+++++++++
UJU
WU
JUIW
UJW
Wmmcoooo=rwm
-m.
bm
mom
cnm
mm
mvcn
bw
m...-~
~o~
-~~
‘ooooooooorl
Nm
$Oooooo
oooc-+om
+++++++++-+-
1l.lwulll.!uu.
lulu.lu+u+o~o--mmou
Omo
VW
WCTILOCOCNO020WCI
t.-m-om+oulowo
m,Qm
uw
mN
mV
omo
..
..
..
..
..
-tU
lullomm
mm
m.
l..
..
..
..
..
..
..
..
..
..-.
..
mbm
mti--
mbw
mm
mm
mm
mm
aw
m
----?yoo
uu
.lLhL
lma-m
VC
Jr-q
03cnww
-$mln
t-,,.
.C
o=rm
cfl.
..
..
..
..
..
..
..
..
C9*M
V*U
)WWWWWWW)
W.-?-W
oo--cN@4m
m**.3Lou)Lnc9law
l.c+
000000000000000000-
++1
8f
18
fI
tt
*I
11
tt
*LLlu.lwu.!luululw
!.uluuluu!$lu!u!ufu!L
C711n
aciwloalnmolm>m.-ocow
ow
U70wu)mf
ncn
wlol
rl.-’-m!m
m>
mm
>.t.
mmmm.
mm
mm
mm
m-h
mfs
;=r-vu-c9ma2*o-mw-t.lD-in
.....................
N-*.
*.w-Fm.
mN
-mm
-qo*o
ill
0zzO
mm
bm
+**
o@
4---------Utih
hm
mm
mm
omm
uciin
mm
ctwm
~=tcv-o
rzm
------U
..
..
..
.
mm
mm
mm
m-----.-
****II
wbm
r40c9
mw
*ofoom
mow
m*Z
rx+ob
vm--
*I-E
C4C-J
-----*O
L1.
1.......
1*11-)Ulln
mu
-)lnlfl
*.-
.-.-,------
*
I
35
u;0:“
Oooooow
(JIOoooooo
noooooom
z.-mm
~m
ow
*******:*++
*L::*Z*2*E**U*J*I-I*L**Z*O**N:10
●Z
**&15*O
nu
olIuolalaGII
0)
o0.0moF10
gjil>u5zuu)
aoaou)
<2>gaU)
gto
JJwv
-1m0c?ne00lxCL
W’s’q;g
Ulw
uw
uu
o-ww
-1-
mO
NO
OI.
WW
m-oln-t
O-
am
ow
p~q;
LLlu
Ju
Jlu
uw
o-llDln
-ln
a@io
-r.
m-m
mm
mo-=
f.
..
..
..
..
..
U-)vm
mew
-rvlnln
ln
000000
yyyyy~
Ww
wu
lww
mm
-m-w
m-rfm
bv
mm
oovmu
-Jc4
@-)-c-4
-..,
!
km”m
mm
m
ooooooo-00000000
+++++++,
Ulu
Jlu
llJ1l.!u
uw
mm
-mo-u
lr-m
mw
omm
mm
..-m-c+
-mm
Cn
Ln
mt-m
r-u-im
..
..
..
..
bemmmmwe
.-.-.-.--,-0
0???y??~
y
UJw
ww
wu
ulu
*mw
m*-w
wm
mm
.tic+
-m.-m
b-trem
or-m
~w
otimbw
..
..
..
..
u?w
19w
t-m--
oo--mm
mm
uu
utn
fflmaw
ot-
N000000000000000000
~+
+1
11
1I
1t
It
I1
t,
,1
,U
Ju
wu
uw
ww
wlu
ulu
lu-lw
ww
uu
.-.-~w
watgm
mm
mm
mm
mom
wk
mm
mm
inm
~=tm
mt-u
obm
ooin.-
om+ow
mm
o--m-c+
am
mom
mm
ciww
omw
m-=
tcomcom
mm
-o*
..—
,..
..
..
..
..
..
..
..
A:w
-w-m
tiam
-ma-w
a-m
emo
mmwg02
S~
O0’0
-000
m000000
mm
~0
--------------
Li$
mm
mm
m---.-
mcom
mm
mm
mm
mm
mfj
,--------------
I-W
wfflw
iil4!3.-
ZC
JW
--m--0
0------0
0**
ao
--~
-~
--
--~
--
--
--
--*
.LL
*V*
W*-+
**---
e-.-,-.
111111
UJw
wu
ulu
lm
ob--m
omr-om
~ow
m+m
-U
ti.1-a
=f
-C+
CJ-0
0
=rQ
lnm
lnw
mu
l-.
T---------
111$1181
WU
ILIU
JU
JU
WW
wm
wm
c40-rr-
tnwbm
-wm
-am
--mb=
rwm
+o-tw
ww
mm
,oomm
so*r-
4.
..
..
..
..
..
..
..
..
+-.-.-
v.y.-.--.
mm
m-m
rn
gm~am
mm
mm
mom
mfnom
mm
****CI
*LXL..
36
RUN TERMINATED 19 SECONOS BEFORE IJOB TIME LIMIT.
SAMPLE PROBLEM FOR MFE
NO PARTICLESUNSCATTERED
TRACKS
\SOURCE 66475SCATTERING oFISSION o \(N,xN) 67FORCEO COLLISION 8535 \
TALKS
LEDGER OF NET NEUTRON CREA”
WEIGHT ENERGY(PER souRcE PARTICLE)
I.0000E+OO 2.5991E+O0o. 0.0. 0.3.6737E-04 9. 1998E-04o. 0.
IoN AND LOSS (FOR ACCOUnting ONLY)
LARGENUMBERLOSTTOENERGYROULE~E TRACKS
ESCAPE
\
62929SCATTERING oCAPTURE 2514ENERGY CUTDFF 5590TIME CUTOFF o
WEIGHT CUTOFF o 1 0. 0. WEIGHT CUTOFF oWEIGHT WINOOW oCELL IMPORTANCE 14468ENERGY IMPDRT. 48832OXTRAN 5EXP. TRANSFORM oOEAO FISSION o
TOTAL 134338
WEIGHT WINOOW
J
o 0. o.’CELL IMPORTANCE 49918 1.1656E-01 1.0066E-01ENERGY IMPORT. 4.6479E-01 1.5012E-01OXTRAN 934: 3.0923E-10 1.2561E-09EXP. TRANSFORM o 0. 0.
TOTAL 134338 1.5817E+O0 2.8508E+O0
PREOICTEO AVG OF SRC FUNCTION ZERO 2.6000E+O0TRACKS PER NEUTRON STARTEO 2.0209E+O0COLLISIONS PER NEUTRON STARTED 6.2437E+O0TOTAL COLLISIONS 415052NET MULTIPLICATION 1.0004E+O0 .0001
COMPUTER TIME SO FAR IN THIS RUN 4.66 MINUTESCOMPUTER TIME IN MCRUN (4CO) 4.61 MINUTESSOURCE PARTICLES PER MINUTE 1.4420E+04FIELO LENGTH 371584 = 13256008RANDOM NUMBERS GENERATED 4653934LAST STARTING RANOOM NUMBER 33054041550251210NEXT STARTING RANOOM NUMBER 7246405510430155B
AVERAGE LIFETIME, SHAKESESCAPE 5.2613E-01CAPTURE 6.7963E-01CAPTURE OR ESCAPE 5.2785E-01ANY TERMINATION 2.5120E+O0
TOTAL NEUTRONS BANKEDPER SOURCE PARTICLE
TOTAL PHOTONS BANKEOPER SOURCE PARTICLE
MAXIMUM NUMBER EVER IN
BANK OVERFLOWS TO DISK
7
S 09/19/83 11: 13:02
WEIGHT(pER SOURCE
6327E-01o.8.6574E-032.3566E-01o.0.0.1.1712E-014.5701E-017.8099E-10o.0.1.5817E+O0
CUTOFFSTCO 1.
ENERGYPARTICLE)
1.6367E+O08.6664E-019.5842E-021.2582E-03o.0.0.1.0112E-011.4924E-018.OI33E-10o.0.2.8508E+O0
0000+123ECO 1.0000E-02Wc’1 o.WC2 o.
618209.2997E-01
o0.
BANK 440
Fig.27. Creationandlossledger—energyrorrlette,nosourcebiasing.
F. SourceEnergyBiasingandEnergyRouletteAppliedto theSampleProblem
Both source energybiasing and energyroulette indi-vidually improved the FOMS.The natural temptationat this point is to try both techniques and hope forimprovement. Before trying both techniques, asuspicious person might wonder whether two energybiasingtechniqueswould be too much of a good thing.Would the calculation be overbiased?Fortunately, forreasons explained below, the techniques work welltogether.
Figure28givesthe resultsof usingboth sourceenergybiasing and energyroulette. First, note that the “tracksentering”column looksvery nice. Second,note that theFOMSare
1. a factor of4 better than energyroulette alone,2. a factor of 2 better than source energybias alone,
and3. a factor of 8 better than with neither energy
roulettenor sourceenergybias.Hindsight, aided by elementary arithmetic (4” 2 = 8)indicates that the two techniques operate essentiallyindependently.AIthough both are energy biasing, theyare biasing different things. Source energy biasing isapplied only at the source and suppliesthe right initialspectrum; thereafter it does nothing to keep the rightspectrumaftercollisions.In contrast, the energyroulettetechnique does nothing to alter the effectsof the initialspectmm.That is, ifN4,~4-MeVsourcetracksproduce atrack distribution nl (r, v, t), biasing the source to in-stead produce N2 14-~~V source tracks wi~ $roduce atrack distribution nz(r, v,t) = (Nz/Nl )nl (r,v,t). Theenergy roulette game takes no account of the sourceenergybiasing.Synergismcan be viewed as follows:thesourceenergybias producesgood initial track distribu-tion on which the energy roulette works to produce agood subsequent track distribution. However, if theinitial track distribution is not good, the subsequenttrack distribution cannot be good because the energyroulette game is independent of the initial track dis-tribution and thereforecannot “correct” it. Energysplit-ting/Russian roulette thus contrasts with the nextenergy-biasingtechnique considered, the space-energy-dependent weight window. The weight window, if setproperly,will correct poor track distributions and if setpoorly,willdestroygood track distributions.
XIII. THE WEIGHTWINDOWTECHNIQUE
The weightwindow(Fig.29)is a space-energy-depen-dent splittingand Russian roulette technique. For eachspace-energyphase-spacecell, the user suppliesa lowerweightbound and an upper weightbound. These weightbounds define a window of acceptable weights. If aparticle is below the lower weight bound, Russian
roulette is played and the particle’s weight is eitherincreased to be within the window, or the particle isterminated. If a particle is above the upper weightbound, the particle is split so that all the split particlesare within the window. No action is taken for particleswithin the window.
Figure 30 is a more detailed picture of the weightwindow. Three important weights define the weightwindowin a space-energycell,
1. W~,the lowerweightbound,2. Ws, the survival weight for particles playing
roulette, and3. Wu, the upper weightbound.
The user specifies (WFN cards) WL for each space-energy cell, and WSand WU are calculated using twoproblem-wideconstants, Cs and Cu (WDWN card), asWs = CsWLand Wu = CuW~.Thus all celIs have anupper weightbound Cu times the lower weight boundand a survivalweightCstimes the lowerweightbound.
A. WeightWindowComparedto GeometrySplitting
Althoughboth weightwindowand geometrysplittingemploy splitting and Russian roulette, there are someimportant differences:
1.
2.
3.
4.
5.
the weight window is space-energy dependent,whereas geometry splitting is only space depen-dent;the weight window discriminates on particleweight before deciding appropriate action,whereas geometry splitting is done regardless ofparticleweight;the weight window works with absolute weightbounds,whereasgeometrysplittingis done on theratioof the importancesacrossa surface;the weight window can be applied at surfaces,collisionsites,or both, whereasgeometrysplittingis appliedonly at surfaces;andthe weightwindowcan control weightfluctuationsintroduced by other biasing techniquesby requir-ing all particles in a cell to have weightWL< W< Wu,whereasthe geometrysplittingwillpreserveany weight fluctuationsbecause it is weight inde-pendent.
B. Special Weight Window Features Described inMK!NPManuall
1.
2.
3.
There is a maximum split/roulette feature thatlimits the amount of splitting/roulettingthat canoccur at any particular weightwindowgame.The window is always adjusted to be at least afactorof 2 wide, that is Wu/WLz 2.A spatial weight window (only one energy range)may be specified inversely proportional to
38
L
Oooooooooooooooooom
mm
Ooooooooooooooooooc.loci
+++-+
+++++
+.+
-++++-+
++-+
-!u
wu
ulu
lulld
uu
wu
lluu
u~u
llul.u
+u
+o-m
mb~
woo-m
ww
oomm
mooo
mm
wN
omm
mu
om*m
mm
mou
o*oW
obam
mom
-mw
wm
mw
wm
moq
om
-mm
mdm
mu
mow
wm
mw
mw
oqo
..
.
ooo-ym
?
L!J
LU
UJLLI
win
mw
CY
mr-o
-mm
mu
rn
-m,..
..-
.,-F
----0000
,,,,
Uluw
wm
-owm
mm
cnm
omri
c+m
r.w
.
,...,
.-.~
.-,-.
..
..
(wC
4m
rtc-i
..,-,----m
yoooo
LU
LU
JJLJIL
II+m
qm
mk
.mm
mm
wci
mw
ww
me
oowom
l-.
..
..
.m
mw
mcir.
l-.-.--,-ev,--.-
y?oooooooo
uklL
IL!L
iJILL
LIL
tLIL
mow
ocnm
vm.-o
ww
mw
cnow
mw
>m
t.-ciw+m
afm
mm
m-m
cccnm
mw
a.,..,
..,.,t-+
wm
mw
oww
w
oo-yy?
WW
wm
loww
mm
m~-
.r.m..
.,-.-m
-mO
-(Dm
-mm
oc9m
~m
mm
ww
ogoO
c4m
mm
bm
mm
wm
+m
-mtn
vmw
mm
mom
tn~bm
oomw
m~m
mN
mw
wm
ww
ww
wr.com
mm
mm
aw
m,-.---..-.-,--------------
~
Aa+o1-
ci-r-0m
c4-m
bO
mr.O
mm
mm
mam
mm
mm
m--m
mm
mom
t-mm
ct~o+m
mw
-omcnm
cumr.t.+
t-E
cxcw.-.-’---ooooooooo
~U
.......
..
..
..
..
..
1-1-am.-
a.
t-W
v
AA
ILA
mm
mm
mm
mci
OW
UU
Im
wm
m.a
,---,,.
.U
ILom
ln
0z
*
P
~+
+tic.r-om
we-rim
r.t.wovcim
-mS
*o-m
m=r=
frmm
m-w
m-+
wm
am
fxmO
bm
m.-0
mfn
mm
mw
wr.p
b~1~
UC
4C
4---.-O
OO
OO
OO
OO
OZ
*C
Ju
..
.,
!.
,.
..
..
..
,.
,-*CO
*<*
=tU
lmm
mu
lmln
mm
mm
mm
mm
lm-*
.--,-.--...--.-.--..p,-
m.
zmm
mci
omm
mm
L
LIL
LL
IIILL
LL
Im
mm
mm
wom
m+r.
mw
wm
vwo
Oqm
o-lwo
-mo
m-o
..
..
..
+r-m
mm
m
Lom
cclwr.t.
mm
mm
mm
*
wm
t-+m
o-m*c4
mr.r.v.-m
mom
-t.wm
w.-m
bq.-+
wm
oorz-m
mr-m
mr4
-Om
mm
mm
mm
mctm
m-----.-oooooooo
w......
..
..
..
..
..
.
mm
mm
m.m
mm
mm
mm
mm
mm
my?oooooo?y?yooooo
UJW
LL
JJLL
LIL
ILL
UW
WW
JJJJLL
LL
LZ
mm
wm
mm
mm
-mw
qmm
q=t.-
*mm
mm
mu)-m
mti-m
omum
m>u
Fm
l.wm
Om
c40m
mw
inrm
vmc.4
-iEm
o=fw
mbow
ww
r.r.obm
mq
-1r-m
t--.-wm
m=f-o=
f---o.
..
..
..-.
..
.,
..
....
:W
fflv=tq
=tvm
ulm
mu
llnfflm
u-lln
0000000000000000+
(nO
oooooooooooooooul
Loooooooooooooooom
z-~m
~m
mbm
mo-c+
mw
maw
-.-----,-,-
39
Fig.29.Theweightwindow.Tracksenteringa phase-spacecellwithweightabovethewindow’supperboundaresplitintoseveraltrackswithinthewindow.ThosewithweightsbelowthewindowplayRussianRoulette.Therefore,particlespassingthroughthewindowhaveweightswithinthewindowbounds.
previously optimized ceil importances from thegeometry-splittingtechnique.
C. Specifyingthe Weight Windows for the SampleProblem
The weightwindowparametersshouldbe such thatthe weightwindowsare inverselyproportionalto thespace-ener~ importance.Thus one must eitherguesswhattheimportancefunctionlookslikeoruseinforma-tion fromexperience.The geometry-splittingoptimiza-tionhasalreadyprovideda spatialimportancefunctionthatcanbe used(see item 3 in Sec.XIII.B)to obtainaspace-onlyweightwindow.If the cell importanceswerenotavailable,one couldeitherpickwindowparametersthat flattenedthe trackdistribution(in the same iter-ative procedureused for geometry splitting)or onecouldusethe weightwindowgeneratordescribedlater.
The weight windows are chosen according to avail-ablecellimportances(exceptfor cells20-22).
W~= 0.5/cellimportances lowerweightboundWs= 3.0. w~ survivalweightWu = 5.0. WL upper weightbound
Furthermore (see item 1 in Sec.XIII.B), no particle (inany givengame)willbe splitmore than five for one, norrouletted harsher than one in five. The weightwindowgamewas turned off in cells20-22because that part ofthe problem is too angle dependent for the weightwindowto be effective.The weightwindowwas appliedboth at collisionsand surfacecrossings.,
D. SpatialWeightWindowResults
The source energy bias and energy rouletteremoved for this run. The following techniquesused:
1 energycutoff,2. forcedcollisionin cell21,3. DXTRAN with DXCPN probabilities,
40
1PARTICLES HERE ~ SpLIT
/UPPER WEIGHT BOUND
SpECIFIED As A CONSTANTC. TIMES WL
PARTICLES WITHINWINDOW =
DO NOTHING
THE CONSTANTS Cu AND C~ARE
FOR THE ENTIRE PROBLEM
-1w~/
SURVIVALAS A
WEIGHT SPECIFIEDCONSTANT
C~TIMES VVL
tLOWER
SPECIFIEDWEIGHT BOUNDFOR EACH
SPACE-ENERGY CELL
PARTICLES HERE a PLAYROULETTE, KILL IOR MOVE TO W~
Fig.30. Detailoftheweightwindow.
41
-1
42
“1‘1“1‘1
“1“11
11
UIW
LU
UU
UJLU
IJ.IU
m.
O=rm
L-m
OW
ecwow
mm
wm
wom
m-m
wow
+ow
w-u
lw-fl~
oooooO
Om
O.
..
..
..
.-,---.--m
-
**O*W5!
*.m
*S**Ill
*c
*!1
*a
*=*+cl)
*U*g
**i-
*~
***1-
*<
*t-
**OC
O*I.
1-
12
~*W*
m***
1!
>-
*-1
*UI
*IJ.
1-*Z*
0*
w**
g*.
n
a‘.
...
,,
,,
..
..
.1-
t--l-ww
aww
wm
~~
000000000000V
looooooooooo+cL
ooooooooooo~zum
cww
O-tm
~wO
*w-.-mmmmm***
CU
*n*l-*Z*3*U**U
.!
Id*L**Z*O**C-I
***.*O*ZIn*Z*3*n
1-incl)
nzawawzv)
4. ringdetector, and5. spatial weight window from refined cell im-
portances.Figure 31 shows the spatial weight window results.
Comparisonwith Fig.23 showsthat the FOM (tally 1)is3 for the weightwindow versus4 for geometrysplitting,but the statisticsare bad on both runs. The main point isthat a spatialweightwindowand geometrysplittinggivecomparable results. In fact, in most cases where thestatisticsare good enough to judge, a spatial window ismarginallysuperiorto geometrysplitting.
XIV. THEWEIGHTWINDOWGENERATOR
The weightwindowgenerator semiautomaticallyob-tains optimized weightwindows.The generator can bevery usefulfor experiencedMonte Carlo users; it is notrecommended for novices. Weight window generatordetails are described in the September 16, 1982, X-6memo, titled “Use of the WeightWindowGenerator.”
A. Comments
1.
2.
3.
4.
The generator requires considerable user under-standingand and interventionto work effectively.The generator is scheduled to become a standardMCNP feature, but is currently only a standard(maintained)patch to MCNP.Running MCNP with the generatortypicallycostsan extra 20-50Y0of the required time for runningMCNP without the generator.Tracking is not affected by the generator; that is,every particleexecutesa random walk identical toits random walkwhen the generatoris not used.
B. ImportanceGeneratorTheory
The importance of a particle at a point P in phase-spaceisequalto the expectedscorea unit weightparticlewill generate. Imagine dividing the phase-spaceinto anumber of phase-space “cells” or regions. The im-portance of a cell can then be defined as the expectedscoregenerated by a unit weightparticle after enteringthe cell. Thus with a little bookkeeping, the cell’sim-portancecan be estimated as
total scorebecauseof particlesImportance (and their progeny)enteringthe ceil(expected =
score) total weightenteringthe cell “
Consider the example of Fig. 32, which represents agenericphase-spacegeometryof four cells.In this exam-ple, the capture probabilityat each collision is ().5,andcapture is treated implicitly by weight reduction inconjunction with a weight cutoff. Particles are born incell 1 and are scored as they leave the slab from cell 4.The S values are used to determine the splitting andRussian roulette games played at boundary crossingsbetween the four phase-spacecells. In practice, these Svalues are usually the user’s best initial guess at animportance function. Each particle trajectory is con-secutivelynumbered. Table III shows the importanceestimationprocessfor the three particlehistoriesof Fig.32. Note also that this importance estimation worksregardless of the variance reduction techniques usedduring the calculation (tracks that reenter the samephase-spacecell should not be counted twice as weightentering).
C. Setting the Weight Windowfrom the EstimatedImportances
Althoughthe generator and weightwindow conceptsare independent, they are complementary. One cannot
TABLE III. ImportanceEstimationProcessforParticleHistoriesin Fig.32.
Row Description Cell 1 Cell2 Cell3 Cell4
Weight1 Trajectoriesentering 1,8,13 3,4,9,10 14,15 6,172 Weightenteringassociatedwithabovetrajectories 1,1,1 0.25,0.25,0.5,0.5 0.5,0.5 0.5,0.53 Totalweightentering 3 1.5 1 1
Score4 Trajectoriesenteringthatresultedin score 7,17 7 17 7,175 Scoresassociatedwithabovetrajectories 0.25,0.5 0.25 0.5 0.25,0.56 Totalscore 0.75 0.25 0.5 0.75
Estimate7 EstimatedimportanceRow6/Row3 0.25 0.167 0.5 0.75
43
VO,ID
CELL 1
s =1
Birth
Birtho 8 ,W=l
W = WEIGHT
s ==SPLITTING PARAMETER
~= LA3ELED TRAJECTORY
o = COLLISION POINT
I
CELL 2
S=2
Weight
cutoffGame
<
5 ~%= 0.25 ~+C))
3
‘= 0..3’
9 w =0.5
CELL 3S=2
CELL 4
S=l
ouletted
‘ouktted
‘oulettecf
17w=fJ.5
Fig. 32. GenericMonte Carlo problem of four cells with three particle histories, illustrat-ing howimportancescanbeestimated.
insist that everv historv contribute the same score (azero variancesolution),but by usinga windowinverselyproportional to the importance, one can insist that themean score from any track in the problem be roughlyconstant. In other words, the window is chosen so thatthe track weight times the mean score (for unit trackweight) is approximately constant. Under these condi-tions, the variance is caused mostly by the variation inthe number of contributing tracks rather than by thevariation in track score.
Thus far, two weightwindow properties remain un-specified, the constant of inverse proportionality andthe width of the window. Empirically, it has been ob-served that an upper weightbound five times the lowerweightbound workswell,but the results are reasonablyinsensitive to this choice anyway. The constant of in-verse proportionality is chosen so that the lowerweightbound in some reference cell is chosen appropriately.For example, in the problem described here, the con-stant was chosen so that the lower weightbound in thesource cell was 0.5. The source particles were of unit
,TallySurface
VOID
weight,so they all started within the (0.5-2.5)window.‘In most instances the constant should be chosen thisway so the sourceparticlesstart within the window.
D. SpatialGeneratorResults
Figure 33 is the same run as ‘Fig.31.except that thegeneratoris turned on. Note that the runs traclcperfectlyand the generator has slowed the calculation by 4%.Typically, the generator will slow the calculation by20-50%,but of course the generator can be turned offwhen a good weightwindow has been generated. Thusno time penalty need be paid for the final run to grindthe statisticsdown.
Figure34 showsthe generatedspatialweightwindowinsertedin the input filefor the nextrun. Manywindowswillbe displayed,so I will explain how to interpret theWFN card entries,lines67-72.Line 67indicatesthat thefirst (and here the only) neutron weightwindow has anupper energy range of 100 MeV. If there were more
44
Oooooooooooooooooom
.mm
Oooooooooooooooooof.-+
om++++++++
+++++++++
+--+
-W
uuuwuw
uuuuuuuwuuu+u+
om
bm
mo
mo
mm
mm
mm
o-o
Fo
uo
Nw
mm
-mam
qm
ramm
flwm
wo
mo
w~oo-m
mm
omw
mw
~tim
moom
oo.m
mo-vw
o.-Nm
-mm
ww
moT
o.
..
..
..
..
..
..
..
..
..
..
mw
mm
vww
wm
mm
mm
mm
mm
m.+
-
-.--~~~~~~~q
q~*~
~w
Fo
o0000000000000000000---,,,,,,,
,,,,,,,,,,,,,,
..
..
..
..
..
..
..
..
..
..
.om
u~bm
-mqw
-m-m
vm-q
vmv
Ll-
Citw
mm
mcn
mom
owim
l.ooo3
Ln
Nm
N@
l-=roL
owfxtioot..wnoawmm
CtullnLn*xTmmmeJ@#Nm
ILI.
..
..
..
..
..
.
CO
+mco
mm
coaalco
mca
.-,------------
oo--~~mm
ww
~mm
mw
ww
t.w
yy??y?ooooyyoooooo-
UU
UW
UW
JJJJL
JJWW
LL
JLL
LL
~mW
-mm
+m
mm
mO
Fm
a3wW
m~
mmcnm-wmc0ma3.-ommqoc4.-
mr-.o-wmmmm-ti-ommmmm
mmmmwv+t.--vmwmqmtmt.
w,,,.,
..,,,.
..........
N-w-m-mwmm-ww-mm-memo
00Itc1mU)
u
..
..
..
..
..
.-tD
lpu
lmln
lnLn
Ln
Ln
ln
Ew
mN
N@
JN
mc9
mm
mcl
ouw.......
..s..
zt-m@
/N
c4cJ
NN
c4c4
mw
0u
a+00,,m
mm
000L
LIL
LJ
!-mm
w-o
OF
C3
Nm
-w
mm
..
.r-w
w
mm
com?
??
$)Luuw
w11-lw
l.oW
mu
lwm
mm
-g::g.
..
.W
ww
w
acorny??
LLlw
wo
mo
Vw
ln-b
inm
mw
:W,
mU
llnui
000000000000W
ooooooooooobcL
ooooooooooot--zw
a~aow
mm
wo-ew
.-.~~~m
mqqq
**m***.******II
**=*i-*C
J*****x*:*O*F
.*h+$U
l*W**:
II*~g
;****I*****I+*C
L
:?*(Y**W*-1*M*U
.
:2*o**W***.*o*z**CL
*Z*3*L3
w1-0z
45
61- 6012 -.001 100 MeV UPPER ENERGY RANGE62- SRCI
~ /\’
o i. E-6 o 2 1.063- SMOOTH BEHAVIORFOR WINDOW64-65-66-
W,ND;; a!!; ;ij!ii;ii~’:::i ::::;;;;EDNEUTRON v%IdGHT7i- 1.6974 E-OG 8. 4~87E -07 4 : O 16; E-07 PREVIOUSRUN72- 0.0000E-02 O. -1. OOOOE+OO73- w\4G 1 2 .05 0 1. E8 O74- WGEN 1 2.01 4 7 10 100
TURN OFF DXTRAN75- FI 2076- F4 21
WHILE 77- F5Y 2005 200 0
OPTIMIZING WINPoo O 19R 1 0 0 WEIGHT WINDOW
*C DXN O 2005 0 100.2 100.2 TURNED OFF IF ENTRY = ODXC”PN”O .001 3R .01 4R
WARNING ..01 .015 .02 .04 .08
VALUES OTHER THAN 0. OR 1. CAN CAUSE TROUBLE .81- .2 .4 1 1 0 00082- FCN 0 19R 1 0 083- EO .01 100
-1 INDICATESZERO84- TO 100 1000 10000 IMPORTANCEREGION85- CUTN 1. OE 123 .01 “O O86- EXTYN o 0-o 17R o 0 0 087- CTME 4588- PRDMP -5 -589- PRINT90-
THIS RUN WILL GENERATEA SPACE-ENERGY WINDOW WITH ENERGY RANGES
o-1 MeV 4-7 MeV
1 - 2.(2-I Mev 7 - 10 Me!!
2.01- 4 MeV 10- 100 MeV
Fig.34. Inputforgenerafigspace-energywindow.
energy ranges, the upper energy bound for the ithwin-dow would be the lower energy bound for the i + 1stwindow.The lowerenergybound for the first windowisalwayszero. Lines68 to 72 are the lowerweightboundsfor cells 1to 23 and read, in order, from left to right andtop to bottom. For example, the lower weightbound incell 15 is 3.5905E-06.A zero lower weightbound turnsoff the weight window and a —1indicates a zero im-portance region where the particle is terminated uponentering.
Earlier,I cautioned that user intervention is required.This intervention can be seen in the WFN1 entries (Fig.34)for cells20-22whereI turned off the weightwindowgameby enteringzerosbecauseI did not want to use thewindowin this highlyangle-dependentpart of the prob-lem. The windowbehaved smoothly,fallingoff roughlyby factors of 2; thus the weight window needed nofurther intervention.
generator was unable to estimate importance for thatspace-energycell because no particle ever left thesespace-energycellsand contributed to tally 1.Note thatthe zero entriesare usuallyfar from the tally surfaceandlow in energy, indicating that low-energyparticles farfrom the tally surface have a hard time tallying, asexpected. If a zero is left as an entry, then no weightwindowgame will be played, an undesirable situation;thus the user must supplynonzero weightwindows.
Figure 37 showshow I adjusted the weightwindows.An adjusted windowentry is indicated by three trailingzeros in the entry. The window was adjusted accordingto two generalpatterns observed from Fig. 36. If Wijisthe lower weight bound in energy region i and spatialcellj, then thesetwo generalpatterns can be expressedasWij< Wi.~,jand Wij< Wi,j.~,wherem and n are positiveintegers.Thus Fig.37wasobtained by interpolationandextrapolationfrom Fig.36.
G. ResultsUsingtheSpace-EnergyWeightWindowE. Generatinga Space-EnergyWeightWindow
As mentioned earlier, the spatial weight window ofFig. 34looks reasonableand is probably about as goodas it willget. Furthermore, experiencehas proved bias-ing in the energydomain to be quite important. There-fore,the generatorwasemployed,usingthe input shownin Fig. 34, to generate aenergyrangeschosenwere
o–l–2.01 –4–7–
space-energy window. The
1 MeV2.01 MeV4 MeV7 MeV
10 MeV10 – 100 MeV
The choiceswerebasedmostlyon experienceand not ondetailed analysisnor on inspiration. In particular, notethat factorsof 2 pervade the Monte Carlo choices.Note(line 79) that DXTRAN has been turned off while aspace-energywindow is generated (C indicates a com-ment card). This is perfectly reasonable because thespace-energywindow willbe used to penetrate the con-crete and will therefore be optimized for tally 1;DXTRAN is used to improve tallies 4 and 5 but nottally 1.
Figure 35 summarizes the run that used the spatialwindow of Fig. 34 to produce a space-energywindow(Fig.36).Note that removing DXTRAN allowed manymore particlesto be run.
F. Discussionof the Generated Space-EnergyWindow
The space-energywindow produced is shown in Fig.36. Wherever a zero entry appears, it means that the
The space-energywindow of Fig. 37 was inserted inthe input file; Fig. 38 shows the results. Tally 1 hasimproved nicely from an FOM of 6 to 43. However,note in the middleof Fig.38that the sourceparticlesarenot starting within the window, indicating that thesourceshouldbe biased so that the sourceparticles startin the weightwindow.
The window (in source cell 2) for 2-MeV particlesis 9 to 45, (recall that the upper bound is 5 timesthe lower bound) (Fig. 37), whereas the window for14-MeV particles is .05 to .25. Recall (Fig. 36) thatprevious source energy biasing gave source weightsof 9.5 and 0.055 at 2 MeV and 14 MeV, respec-tively. From this lucky coincidencewe already knowthe proper source biasing. Without this coincidence,one could experiment with different source energy bi-asing until the last column of Fig. 36 indicated sourceweights within the window.
H. Results Using Space-EnergyWindowand SourceEnergyBias
Figure 39 shows the effect of starting the sourceparticles within the window; the FOM for tally 1 im-provesfrom 43 (Fig.38)to 75.The onlypeculiarthing inFig.39is the suddenriseand fallin the “tracks entering”and “population” columns around cells 6 and 7. A re-examination (see Fig. 37) of the adjusted space-energywindow reveals that the window for cell 6 in the sixthenergy range looks wrong; it does not tit the generaIpattern. This entry was altered from 3.4489E-04 to2.2000E-3.Also, the window for cell 16 in the secondenergyrange was altered from 4.5208E-6 to 1.0000E-5.Although cell 16’swindow was not responsible for the
47
~ooooooooooooooo
LEO
oooooooooooomm
oooooooooooooo~~
!xOoooooooooooom
ln~oO
OO
OO
OO
GO
OO
O-P
..
..
..
..
..
..
..
ocooooooo----~
~$$
:2~
mw
LL
1.Ld
JLIJIL
LIJ
*z
r-r--m
-m-J
mclm
rom>:
mm
omam
~-J
Zm
.mm
mw
m-1
CJlt-ln
uo
loz
a.
..
..
..
..
..
..
..
1-
Qooooooo
o------mw
..
..
..
..
..
..
..
..
..
..
.m
m_-u
-L9-
mm
-mm
-mm
-m
mm
o
00-
OQOOOOOOOOOOOOO
00
aG0000000000000
co
c’00
++,+++++
+7++++++++++
ww
ww
uw
ww
ww
ww
l-uu
u!wIL
ww
wM
O-
P-
m!t-
L?
-X
lN
bM
mIT
WO
uO
C*Jb
CC
7T
CL-l_
-d
-M
U,
W0>
Go
UP
oG
,N
=r.
m@
oLo
m@
LO
m(w
rm
mr
c“r
c=
LX
mo
t-o
-m
~I.
mm
o0
00
0c,
o-
0m
..
..
..
..
..
..
..
..
..
+II
Uz~
ooooooooooooooop
u
Nci
.---.--.-
.-.--.=.
-?--..-.-0
0????
000000
0000000000
,,
!t
,,
,,
II
t!
,t++
LU
UJW
UJ
UIL
IUW
WLLIW
LU
LU
WW
UW
WU
JW
ur.
01m
*o
mw
-C
4-
mm
nu
me
com
0W
mooa.
wm
mco
mt-
wau
mu
)no
l-touo-1-o~oo
U-Ju
-)ut.m
lnu
c’)-om
oom--~
mm
mv
bm
oa-csL
or.
vLo
Ciooooooooooooooo
0000000000
-----*C
looooooooou
lafm
aaw
Eooooooooo+
+l-1
-awLo
.......,....,,.oooooo
o--
..
..
..
..
.ai~
”m’i
&A
m”v’u
iiiiwm
ww
ww
-wo
mm
mm
Lo
mm
-n.--.-----
-a
z>:
AZ
1a1-
00000
,,,,,1
LLJIW
UJU
IWW
W.C
41.
l.om
aw
vmm
mm
mm
mcl
mv
190
mm
0o3
m@
Jor-oo*
bm
-cxt-v-o
..
..
..
..
..
In“>
Z+ci
Ozo
-0+In
I+ln-u
.-!3
1-1O
*!Yu
&
20.m.-!Jovz0H
oo--~~mm-
w=fm
mw
ww
b+
N000000000300000000
-++
aI
t$
11“
t1
I1
$I
11.t
II
IJww
uu
.lw
ww
ww
ww
wu
luw
wld
wm
-r-av-m
eow-m
mu
mo-~
F0
wm
LO F
m-----mmww0-mmm
-C3MWmOI-0
+m-0
mnoummt.
mm~mm-mobmm-m-mmor~
w.
..
..
..
..
..
..
..
..
..
..
N.v-m
-m-wm-uwmu
amm
o-o
IAW)
&cfm
bm
txmw
mm
omw
mbm
-X
omm
mm
wm
m--m
ma-m
w.xrim
ooo-mm
mm
a-O
ma
Oa
E1.l.nln.7*c9cwcwcNw@la
@J--
ZoUP..
..
..
..
..
..
..
.<s
omwm1-
oooooooooooooo~
Lnc)ooooooooooooo
-aooooooooo@oooo
-zcd
vam
om=tw
mom
vwm
m.-,-
..-.-N
NC
44
CJ
C.4
48
Fig. 36. Space-energyweightwindowproduced.
WFN i 1. OOOOE+OO-1.0000E+OO 2.6000E+01 2 .6000E+01
9. 6000E -01 3. 2000E -01 1. 1000 E-O 13.9000E-03 !.3000E-03 4.4000E-041.6000E-05 5.4000E-06 i.8000E-06o. -1.0000E+OO
WFN 2 2.0100E+OOO’-1.0000E+OO 9.0000E+oo 4.5000E+O0
5.6000E-01 2.8000E-01 1.4000E-011.2125E-02 5.0272E-03 7.6270E-044.5208E-06 2.5657E-06 7.6579E-07o. 0. -1.0000E+OO
WFN 3 4.0000E+OO-1.0000E+oo 3.0000E-01 1.1223E-018.1246E-03 4.7221E-03 2.0442E-031.5468E-04 6.2149E-05 2.1354E-051.8644E-06 8.4843E-07 3.9794E-07o. 0. -1.0000E+OO
WFN 4 7.0000E+OO-1.0000E+OO 5.0000E-02 i.6534E-02
2.9056E-03 9.7180E-04 5.0000E-045.1451E-05 2.5000E-05 1.1916E-051.1805E-06 6.2537E-07 3.7240E-07
8.6000E+O03.5000E-021.5000E-046.0000E-07
2.3000E+O07.0000E-021.6000E-042.7840E-07
4.8144E-027. I146E-049.6969E-062.0447E-07
1.12S5E-022.5524E-045.0000E-06i.9655E-07
o. 0.WFN 5 I.0000E+OI
-1.0000E+OO
-1.0000E+OO 5.0000E-02 1.1635E-022.3599E-03 1.0721E-03 4.0000E-043.8697E-05 2. I199E-05 1.0413E-051.5106E-06 5.0000E-07 2.0000E-07
4.1318E-031.4119E-045.0000E-06f.0000E-07
o. 0.WFN 6 1.0000E+02
-1.0000E+OO
-1.0000E+OO 5.00CK)E-02 1.3488E-023.4489E-04 1.1949E-03 5.0000E-045.7428E-05 2.0000E-05 1.0432E-051.2537E-06 7.1526E-07 2.6978E-07o. 0. -1.0000E+OO
1.0148E-022.1723E-044.6621E-061.0000E-07
WFN 1 I.0000E+OO-1.0000E+OO O. 0. 0. 00. 0. 0. 0. 0.0. 0. 0. 0. 1.4100E-048. 1682E-05 7.4357E-06 1.6792E-06 6.R234E-07 O.0. 0.
WFN 2 2.OtOOE+OO-1.0000E+oo O.
0. 0.
-1.0000E+oo
o. 0.0. 0.
0.0.
1.2i25E-02 5.0272E-03 7.6270E-04 1 .0793E-04 3.9958E-054.5208E-06 2.5657E-06 .7.6579E-07 2.7840E-07 O.0.. 0. -1.0000E+oO
WFN 3 i.0000E+oo-1.0000E+oo O. 1.1223E-01 4.8144E-02 2.4881E-02
8.1246E-03 4.7221E-03 2.0442E-03 7.1146E-04 3.9582E-041.5468E-04 6.2149E-05 2. 1354E-05 Q,5~6gE-06 4.0856E-061.8644E-06 8.4843E-07 3.9794E-07 2.0447E-07 O.0. 0. -i.000oE+oo
WFN 4 7.0000E+’00-I,0000E+oo O.
2.9056E-03 9.71SOE-045.1451E-05 9.6495E-061.1805E-06 6.2537E-07o. 0.
WFN 5 i.0000E+Ol-1.0000E+OO O.
2.3599E-03 1 .0721E-033.8697E-05 2.i19!2E-051.5106E-O6 2.6978E-07o. 0.
WFN 6 1.0000E+02-1.0000E+OO 5.0000E-02
3.4489E-04 1. 1949E-035.7428E-05 2.846iE-0~1.2537E-06 7.1526E-07o. 0.
1.6534E-022. I018E-c)41.19i6E-053.7240E-07
-i .0000E+OO
1. i635E-021.5395E-041.0413E-051.3489E-07
-i .OC)OOE+OO
1.3488E-026.2029E-051.O.$32E-052.6978E-07
-1.0000E+oo
1. 1285E-022.5524E-04
,3.8168E-061.9655E-07
4. 1318E-031.4119E-045.3956E-071.8884E-07
5.3483E-031.3185E-042.4425E-06o.
3.4173E-037.6465E-052.6293E-06o.
1.0t48E-022. 1723E-044.6621E-062.1982E-07
4.7609E-038.2253E-052.2761E-06o.
2.9000E+O0i.2000E-024.9000E-05o.
I.1000E+OO3.5000E-023.9958E-05o.
2.4881E-023.9582E-044.0856E-06o.
5.3483E-031.3185E-042.4425E-06o.
3.4173E-037.6465E-052.6293E-06o.
4.7609E-038.2253E-052.2761E-06o.
Fig.37. Adjusted(byhand)space-energyweightwindow.LoOkfOrthreezerosasindica-tionofhandadjustment.
49
I
SOURCESCATTERINGFISSION(N,XN)FORCEO COLLISIONWEIGHT CUTOFFWEIGHT WINOOWCELL IMPORTANCEENERGY IMPORT.OXTRANEXP. TRANSFORM
TOTAL
LEOGER OF NET NEUTRON CREATION AND LOSS (FOR ACCOUNTING oNLy)
TRACKS WEIGHT ENERGY TRACKS WEIGHT ENERGY(PER SOURCE PARTICLE) (PER SOURCE PARTICLE)
79266 1.0000E+OO 2.6006E+O0o 0. 0.0 0. 0.
428 3.4146E-04 7.8500E-0430 0. 0.
0 0. 0.95295 9.4426E-01 1.18i8E+oQ
\
o 0. 0.0 0. 0.0 0. 0.0 0. 0.
,750,9 \,.9446E+m 3.7831E+O0
ESCAPESCATTERINGCAPTUREENERGY CUTOFFTIME CUTOFFWEIGHT CUTOFFWEIGHT WINOOWCELL IMPORTANCEENERGY IMPORT.OXTRANEXP. TRANSFORMOEAO FISSION
TOTAL
50305 7.6849E-Oi 1.6402E+O0o 0. EI.7143E-01
18796 9. II07E-03 9.5665E-026783 2.2693E-Ot 1.25i2E-03
o 0. 0.0 0. 0.
99135
\
9.3807E-01 1.1746E+O0o 0. 0.0 0. 0.0 0. 0.0 0. 0.0
i75049
PREDICTEO AVG OF SRC FUNCTION ZERO 2.6000E+O0
\
AVERAGE LIFETIME, SHAKESTRACKS PER NEUTRON STARTEO 2.2060E+O0 ESCAPE 5. IIOGE-01COLLISIONS PER NEUTRON STARTEO 5.4481E+O0 cAPTURE 9.3320E-01TOTAL COLLISIONS 43i852NET MULTIPLICATION
CAPTURE OR ESCAPE 5.t60fE-Ot1.0003E+O0 .0001 ANY TERMINATION 1.5297E+O0
\
o. 0.i.9446E+O0 3.7831E+O0
cUTOFFSTCO 1.0000+123ECO 1.00WE-02Wcl o.WC2 o.
COMPUTER TIME SO FAR IN THIS RUN 4.67 MINUTESCOMPUTER TIME IN MCRUN (4CO) 4.61 MINUTESSOURCE PARTICLES PER MINUTE 1.7i98E+04FIELO LENGTH 376688 = 1337560BRANOOM NUMBERS GENERATED 4305494LAST STARTING RANOOM NUMBER 0527527715031145BNEXT STARTING RANOOM NUMBER 603270047t631661B
3967 SOURCE PARTICLES HAO WEIGHT ABOVE WINOOW.BY INCREASING WW ENERGY INTERVAL: o 0 0 0 0 ~WINDOWIDOING WHAT SOURCE3967
75299 SOURCE PARTICLES HAO WEIGHT BELOW WINOOW.BIASING OUGHT TO BE DOING
BY INCREASING WW ENERGY INTERVAL: o 75299
r NOTE IMPROVEMENT FROM FOM = 6 AND LOW MEAN
TIME ‘4.61 MIN
NPS40008000
1200016000200002400028000
360004ooca44000480005200+3560co600006400066000720007600079266
TALLY 1MEAN
4.26255E-083.93201E-063.62294E-083.78523E-083.84399E-083.69001E-083.74802E-083.96024E-083.95712E-063.96848E-064.01664E-084.06935E-064.31095E-084.41742E-084.45244E-064.44663E-084.42566E-084.50210E-084.46087E-084.48032E-08
ERROR.3126.2170.183i.153i.1436.1348.1250.1155.1076
1022:0973.0922.0876.0837.0808.0790.0764.0742.0720.0704
FOM”4044454843424242434343434343434343434343
TALLY 4MEAN ERROR
o. 0.00001.BO090E-15 .99993.75877E-t5 .75162.8190B,E-15 .75182.25526E-15 .75161.87939E-i5 .7518i.6i090E-15 .75182.62570E-15 .52103.35794E-15 .42303.022t4E-15 .42303.62i70E-i5 .40153.31989E-t5 .40153.66146E-i5 .37354.826iiE-15 .33944.50437E-15 .33945.16357E-15 .30746.64840E-15 .27798.14304E-15 .24658.58P57E-15 .22968.23565E-15 .2296
TALLY 5FOM MEAN ERROR
o 0. 0.00002 1.25350E-16 .99992 8.35669E-19 1.00002 6.26752E-19 1.0000i 5.01402E-t9 1.0000i 4.17835E-i9 f.00001 3.58144E-19 1.00002 1.65752E-18 ;63402 1.7363LIE-18 .55362 1.56274E-i8 .55362 1.80765E-18 .48492 1.65720E-i8 .48492 1.93062E-18 ;43672 2.66350E-18 .39212 2.48593E-18 ~382i2 2.64505E-18 .34583 3.37077E-18 .3i063 4.31OOOE-I8 .27384 4.42186E-18 .25864 4.23966E-18 .2586
FOMo211000111f1t1122333
TIME ‘4.61 MIN
Fig. 38. Sp8ce-energywindOw.
50
gwwmmcw-.-.-...----ooooo
L
51
peculiarity,10-5just lookedmore reasonablebecauseinenergyrange 2, the windowswere decreasingby factorsof 4. Figure40 showsthe correctedwindow.
Figure 41 shows the results of correcting the badwindow. The “tracks entering” and “population” col-umns lookmuch better. Perversely,the FOM decreases,but the decrease is not statistically significantand thecorrectedwindowwasused for subsequentruns.
I. ExponentialSourceAngle Biasing and the WeightWindow
Recall that exponential source angle biasing did notimprove the FOMS for the problem. As with mostbiasingtechniques,competingfactorsaffectcalculation.Exponentialsourceanglebiasingpreferentiallysamplessourceneutrons moving closeto the+? direction. Thussource neutrons that are more likely to score are sam-
pled more often.However, the biasingalso introducesaweight fluctuation that the geometry splitting/Russianroulette technique preserves. Probably the negative ef-
fects of this weight fluctuation cancelled the benefit ofsampling more important source neutrons more often.
A conferenceparticipant (John Hendricks, Los Ala-mos)suggestedthat the exponentialsourceanglebiasingmighthaveworkedif it hadbeentriedwith the weightwindow technique rather than with geometry split-ting/Russianroulette.He said that the weightwindowwould probablyalleviatethe weight fluctuationprob-lem;thusthe exponentialsourceanglebiasing,in con-junctionwith the weightwindow,probablywould ire-.provetheFOMS.
I agree with his assessment and ‘inclucleit here,without proofl as a good example of analyzing theinteraction of different variance reduction techniques.However, the source angle biasing should not be ex-pected to yield the same dramatic improvement inFOM as the sourceenergybiasbecausethe particlesthattaI1ywilltypicallyhave many collisionsand wiIlquicklyforget their source angle. In other words, after a fewcollisions,a preferred source particle willbe essentiallyidentical (exceptpossiblyits weight) to an unpreferred
WFN 1 1. OOOOE+OO-1. OOOOE+OO 2. 6000E+OI
9. 6000E -01 3. 2000E -013. 9000E-03 1. 3000E -031. 6000E-05 .5. 4000E-06o. 0.
WFN 2 2. OIOOE+OO-1. OOOOE+OO 9. OOOOE+OO
5. 6000 E-O 1 2. 8000 E-O 11. 2125E-02 ,5 .0272E-03
ALTERED FROM 4.520SE-06 ~ 1. OOOOE-05 2. 5657E-06o. 0.
WFN 3 4. OOOOE+OO-1. OOOOE+OO 3. OOOOE-Ot
S . 1246E-03 4.7221 E-031. 5468E-04 6.2 149E-051. 8644E-06 8. 4843E-07o.
WFN 4 7. OOOOE+OOO.-1. OOOOE+OO 5. OOOOE-02
2. 9056E-03 9. 7180E-045.1451 E-05 2. 5000E-051. 1805E-06 6. 2537E-07o. 0.
WFN 5 1. OOOOE+OI-1. OOOOE+OO 5 .0000E-02
2. 3599E -03 i .0721 E-033. 8697E-05 2. i i99E-051.5 I06E-06 5 .0000E-07o. 0.
WFN 6 1. OOOOE+02- i . OOOOE+OO 5. OOOOE-02
ALTERED FROM 3.4489E-04 ~
/
2. 2000E-03 f . 1949E -035. 7428E-05 2.0000E-051. 2537E-06 7. 1526E-07
/ o. 0.
/
2 .6000E+011. 1000E-014. 4000E -04i . 8000E -06
- i . OOOOE+OO
4. 5000E+O01. 4000E -Of7. 6270E-047. 6579E-07
-1. OOOOE+OO
$. f223E-012. 0442E -032. 1354E -053. 9794E -07
-1. OOOOE+OO
1. 6534E -025. OOOOE-041.19 i 6E -053. 7240E -07
-1. OOOOE+OO
1. 1635E-024. OOOOE-041.04 i 3E -052. OOOOE-07
-1. OOOOE+OO
1. 3488E -025. OOOOE-041. 0432E-052. 6978E -07
-1. OOOOE+OO
8. 6000E+O03. 5000E -021. 5000E -046. OOOOE-07
2. 9000E+O01. 2000E -024. 9000E -050.
2. 3000E+O07. OOOOE-021. 6000E -042. 7840E -07
4.8 144E -027.1 146E -049. 6969E -062. 0447E -07
t . 1285E -022. 5524E -045. OOOOE-061. 9655E -07
4.13 18E -03i .41 19E -045. OOOOE-061. OOOOE-07
1 .0148E -022. 1723E -044.6621 E-06i . OOOOE-07
I.1000E+OO3.5000E-023.9958E-05o.
2.4881E-023.9582E-044.0856E-06o.
5.3483E-031.3185E-042.4425E-06o.
3.4173E-037.6465E-052.6293E-06o.
4.7609E-038.2253E-052.2761E-06o.
52
‘~ llllSISEXPLANAT!ON FOR STRAN6iE
Fig.4& Adjusted (byhand) space-energy weightwindow.
BEHAVICJR INTRACKSENTER]W
0000000000000
0000000000000
+++++.+
++-+
++++
u.llu
ulllllu
luu
lll.luu
u~~
Lm
c9t.w
ul.oN
*Ln
c.lqow
vFcn
lnm
rmr.o+
mm
~w
m-o.m
+.m
a.m
~O
o.nao-l.m
mco.(-ocfjm
1-
1-
(AZ1-z
010
l-fc3l-
wl.m
HU
H_J
>z
-1O*IY
Uu
.!C
L
20HVIu-1
A-1wv
Ooooom
OO
OO
ON
+-+
i..i-+.-
LLlu
fuu.lu
+(m
moo.o
Co.-ln
l.mo
Coovw
mo
Vmmmwo
......
Inzm
-N.N
tv.sJm
PN
@J.-vr..-~
N0L
.-.-,-????Iu
ulu
wC
n-vln
mow
mLo(D
cn
Nm
mcw
m,.
.-.
‘$$$:$:
UIU
UU
IUJ
!J$7)C
rlulou
lul
-mm
bw
mC
nom
*!nl-
t-!ncn
wm
w.
..
..
.W
wavm
ci
M*
m
U)u
l-.-
Ui!n
ul
r---
liLL
l!J‘m
w’-
m-o
mln
-m
cocgO
wo
..,,.F
min
mm
U)ln
lnm
mlf)
.,-.-er
IIIltt
Uu
lwu
u.lu
,lO
mvm
lnoInw
cn!.ocg.Nt+m@Job
~E
%%
z%.
..
..
.
//
53
source particle. In contrast, a 14-MeVsource particlewill typically have higher energy in every part of theproblemthan a 2-MeVsourc,eparticlewouldhave. Thusthe benefitof sourceenergybiasingis feltthroughout theentire problem, but the source anglebiasingwill be feltonly within a few free paths of the source. Most of thesampleproblem is more thah a fewfree paths from thesource, so I would be surprised to see more than a 10%FOM improvement with any type of sourceanglebias.
XV. THE EXPONENTIALTRANSFORM
The exponential transform in MCNP stretches dis-tances between collisionsin the forward direction andshrinks them in the backward direction by modifiingthe total macroscopiccross sectionby
where p is the cosine of the angle with respect to areferencedirection (currentlyonly +~ in MCNP) and pis the user input exponential transform parameter(O< p s 1)with
p=() no biasp= 1 completebias. !
Many claims for the exponential transform exist inthe MonteCarlo literature,but they are usuallybased onanalysisofone-dimensionalproblemsand often on one-dimensional monoenergetic problems. In practice atLos Alamos, the exponenti~ transform is considered adangerous biasing technique unless accompanied byweight control (for example, the weight window inMCNP), In fact, so many MCNP users had problemsobtainingreliablemean and variance estimateswith theexponential transform (when used without the weightwindow) that the technique was sometimes referred toas the “dial an answertechnique.”
LosAlamosexperienceindicatesthat the weightwin-dow eliminates the “dial an answer” phenomenon andthat the exponential transform can be effective whenused with a weightwindow.The exponential transformboth with and without a weightwindowwillbe demon-strated on the sampleproblem.
A. Comments
1.
2.
54
MCNP givesa warningmessageif the exponentialtransform is used and a weightwindowis not.The exponential transform is not recommendedfor novices.
3. The exponential transfrom works best in highlyabsorbingmedia and very poorlyin highlyscatter-ing media.
4. Empirically,p = 0.7 seemsto work wellfor shield-ing calculations on fission or fusion spectrumswith shieldingmaterials lilceconcreteor earth.
5. There is a standard (maintained) patch to allowthe referencedirection to be arbitrary, not just+?a$currently implemented.
B. The Sampleproblemwith the ExponentialTrans-form
Anexponentialtransform (withp = 0.7)wasadded tothe input file that produced Fig. 42. That is, the follow-ingtechniqueswereused in the next run:
1.2.3.4.5.6.
energycutoff,forcedcollisionin cell21,ring detector,space-energyweightwindow,sourceenergybiasing,and.exponentialtransform (p= 0.7)
Figure 42 shows the results of using the exponentialtransform with a space-energywindow; the FOM im-proved from 72 to 126.Results from running the sameproblemwithout the space-energywindoware shown inFig. 43. Note that the errors are much worse, andmoreover, are not decreasing monotonically with in-creasinghistories. Admittedly, the error levels are toohigh to make them reliable;however, one can certainlyexceptlessjumpy statistics.For instance, compare talIy1 with tally 4 of the previous table; Note that eventhough the initial errors are high for tally 4, they aredecreasingmonotonically.Jumps in the relative errorsindicate a few largeweightparticles trouncing the tallyand thus indicate poor sampling. I have seen suchrelative error jumps frequently at the IOOklevel andoccasionallyat the 5%level. The higher the.transformparameter is and the more collisionsthat are undergoneper particle,the worse thesejumps become.The weightwindowsplitsparticlesbeforetheir weightscan becomeexcessiveenoughto trounce the tallies.
Concerning tally 1 of Fig. 43, note that at 80,000histories,the stated resultsare 2.75E-8& 30.6%,yet Fig.42 indicates that the true mean is close to 4.85E-8.Aquickcalculationgives
standard deviation = .306”2.75E-8 = 8.41E-9“true” - estimate= 4.85E-8- 2.75E-8 = 2.lE-8standard deviations 2.lE-8 z 5from the true mean = m= “ratio true/estimate = 1.76,
which indicatesjust how unreliableerror estimates canbe whenthe samplingis poor.
Lom
tnw
y??y
WU
LU
LU
InLn
ou
-1m
m-o
N.0
00,
mo
m.
..
w-b
arn
??13??
Ww
ww
uW
VII-)F
UY
mm
owr.
WO
W=JN
mm
Cw
l--.
..
..
-mm
mm
.,.,,
0000
yyyy
Luullu
luC
oc
Na
l*lo
v-om
mcn
mm
wm
m,..
, ‘00000
‘yymyy
UL
UU
.!UJ
UJ
W’u
--oO
WW
I(9Ft-
Lm
@Jm
NW
WV
CJN
..
..
.
0000
yyyy
L1.!u
u.
lluLn
wm
w-m
ooO
t--wm----
....
..
..
..
,.
..
.,
.,
...
.>
.-,
C4
.m
l-v.v.in
No-llnN
-lnNrvo*o
1-
1=-m
u)
Wcocom
a)mm
mcocom
m-
.-,---------------
coco
maam
--------
..
..
..
ulm
mw
=ru
LL
ILIL
JJA
A+w
coc.m
mL
OI.
WW
FN
11-
JWw
lvr.clcim
@4
m--
mfflo+
ma
..
..
..
Ww
w-$
ww
vwT
-.-,--,------
LL
LIJJL
LIL
JJJJILm
mco.
c4m
-mW
ul-o+
o.mw
mm
m-m
mcw
mO
wm
oo-om-o
-N----
,...,...
JIILJJA
LL
LIL
Im
mw
mw
m.-
mt.
lnr-m
l-r.m
mm
-wm
mr.mmmwom
000+000
.......
1-.-m
~.e.--m.-..---m
-py
**o*w*.******II
1=++~
:*****I*:;*o*-*CO
●*:II
*:2*z*******:Ilu*O.
*l-*Z*2*WIW*J*!+*L**Z*O**N***.*O*Z**O
.*Z:2
C-i
*..-255
1-aX
*G31=-lo
wtL
1-lz.LLlu
l3
mz0l-lml-l-1-10vz0l-l1-a-13n
.0n
-1m0IYJnJU
Jn!U
u0En
,...,.
alt--f9r-c-t-
go
oo
oo
oo
oo
oo
ou
000N
NN
111
WU
ILL
Im
mw
Ow
co.-C
70w
tim-m
m4
,,....
..
..
..
1-o
oO
OO
OO
OO
fnam
~oo
oo
oo
oo
oo
oo
LL
Ww
w.
..
56
zU.1
A*
1-A
‘K**8
**O*W*.*W****II**Z*l-*V***********t-*o
)*m*O*G
3
:**II**U
Y*C
L*Z****:***U
I*n
.*l-*Z*3
*U**W*-I*I+*L**Z
”*O**C
+it**.*O*Z**n
.*E
*3*O
IJJ>az
(4v&.-LE
l
XVL THEGRANDFINALE—TURNINGDXTRANBACKON
Recallthat DXTRAN was turned offwhilethe space-energy window and the exponential transform op-timized the penetration. Figure 44 shows the results ofturning DXTRAN back on. This “best” run uses
1. energycutoff,2. forcedcollisionin cell21,3. ring detector,4. space-energyweightwindow,5. sourceenergybiasing,6. exponentialtransform (p = 0.7),and7. DXTRAN with DXCPN card.
As observed previously, DXTRAN vastly improvestallies4 and 5 at some expenseto tally 1.
XVII. CORRELATEDSAMPLINGANDPERTURBATIONCAPABILITY
A standard MCNP perturbation patch allows up tothree slightlydifferentMonte Carlo problems to be runsimultaneously.The perturbation calculationestimatesthe difference in tallies between similar Monte Carloproblemsand it estimates the standard tallies.*
Another way of estimatingperturbationsis correlatedsampling in MCNP that allows tally differences to beestimated between two different runs by correlatingtheir random number sequences.The iti particle in run#2 is started with the same random number that starts
the fh particlein run #1. Becausetheithparticlein run#1might use kl random numbers, and the dhparticle in run#2 might use k2 # kl, random numbers, the i + 1stparticledoesnot startwiththe nextrandom number inthe sequenceafter the ithparticleterminates. Instead, thei + 1stparticlestarts withtheJthrandom number beyondthe starting random number for the fh random number.In other words, there is a random number jump of Jrandom numbers between the start of particle i and thestart of particlei + 1.Thus the ithparticle in runs #1 and#2 will both be starting at the(i–1) . J position in the random number sequence.J, ofcourse,shouldbe largeenoughso that both kl and k2areless than J for all particle histories. This correlation ofrandom number sequencesis depicted in Table IV.
*Forfurtherinformation,refer to video reel #24,“VariousMCNPPatches,ColumnInput,ExponentialTransform,Im-portanceGenerator,Perturbation,”by RobertG. Schrandt,from MCNPWorkshop,LosAlamosNationalLaboratory,October4-7,1983.AvailablefromRadiationShieldingInfor-mationCenter,Oak RidgeNationalLaborato~,Oak Ridge,TN37830.
The correlated sampling problem is identical to thesampleproblemexcept that the densityin cell 21 (Fig.3)hasbeen changed.The two correlatedproblemshave
1. densityin cell 21 = 2.03E-4and2. densityin cell21 increasedby 1%to 2.0503E-4.
Figure 45 summarizes the two problems, each run for20,000histories. Everythingis identicalbetweenthe twosummary charts up to cell 21 becauseall particles haveexactlythe same random walk until they enter that cell.Furthermore, a particle entering cell 21, where the ran-dom walks diverge, will probably never scatter backtoward cell 19. Presumably, if enough particles wererun, backscatter from cell 21 would cause very smalldifferencesin cells 1-19.
Figure 46 shows FOM tables for the two problems.Note that the means differ by about IYoand that thisdifference appears to be statistically insignificant be-cause of the 9Y0errors in the means. However, thesecharts can be used to obtain batch statistics on 20batches of 1000. That is to say, the numbers can bepostprocessed to figure out the tally for each batch of1000particles and then the differencein tally for eachbatch of 1000 particles can be computed. Error esti-mates in the tally differencecan then be made on thebasis of the 20 tally differences.Figure 47 showsthe 20meansfor each problemand the mean and relativeerrorof the difference.Note that with correlated sampling,al~o difference has been found to within 8Y0despitea 9~0error in each of the problems.
XVIII. PHOTONS
The sampleproblem describedhere is a neutron-onlyproblem. Regarding the variance reduction techniquesin MCNP, whatever can be done for neutrons can bedone for photons. Only the neutron-induced gammaproblem needs special consideration. The difficultyarises in setting reasonable parameters (PWT card) todecide when a photon should be produced at a neutroncollision. These parameters specifi, on a cell-by-cellbasis, the minimum weight for producing a photon.This weight should be inversely proportional to thecells’ photon importance. One either has to make aguess or obtain an adjoint solution, such as provided bythe weight window generator. In fact, ifa photon weightwindow is used, these (PWT) parameters should bechosen as the lower weight bounds for the most impor-tant particles (typically the highest-energy window).
XIX. FUTUREPLANS
Goals for the future are1. more automatic biasing (learning techniques),
57
2.
3.
4.
xx.
TABLE IV. RandomNumberUsage in CorrelatedRuns
RandomNumbersforRun#l RandomNumbersforRun#2(*indicatestherandomnumber (*indicatestherandommumber
wasactuallyused) wasactuallyused)
0.14784 * firstparticlestartshere 0.14784 *0.29376 * 0.29376 *-. --— .0.21632 * 0.21632 ‘0.78048 0.78048 *0.14336 0.14336 *0.10304 0.103040.66592 0.665920,38144 * secondparticlestartshere 0.38144 *0.52416 * 0.52416 *0.22912 * 0.22912 *0.03968 0.039680.15776 0.157760.14464 0.144640.25248 0.252480.46272 * thirdparticlestartshere 0.46272 *0.75904 * 0.75904
weight window and generator in more arbitraryphase space,several random number generators (tallies shouldnot affect random walks, and mode 1 neutronsshould track mode Oneutrons), andmore perturbation capability.
CONCLUSION ~
The Los AlamosMonte Carloneutron/photonpar-ticle transportcode, MCNP, containsmany effectivevariancereductioncapabilities.However, these tech-
58
niques must be used judiciously and their effects mustbe monitored using the summary information providedby a Monte Carlo run. This paper has illustrated most ofthe MCNP variance reduction techniques on a concep-tually simple, yet computationally demanding, neutrontransport problem. These illustrations should help nov-ice users better understand the capabilities of MCNPtechniques more concretely than presented in theMCNP manual, which I hope this report will comple-ment. Whereas the MCNP manual must be completeand general, this report makes no attempt to be either.Use this report to get a flavor for MCNP and the manualto setup problems.
-10uz02
00
00Ii&
r-mm
-U
lmm
-
r-r.
00000
‘00000
+++++
UU
JWU
JUN
Ln
mm
oC
Oom
oul
v-enC
9t-
Tv(n
mv
..
..
.m
m.w
mm
s88g$
+++++U
IWU
JWW
Nln
cslmm
Cim
Ln
mol
o-mm
m-m
mF
m
&C
irn&
li
O-mwcl
yc))y~
Wu+
w+
000+0
Pmoii
Jo
mr+
o-oI.
Ooln
o.
..
..
m.--
h-
Wu
l!-t.om
?yoo--
,,,,
UU
UIW
UJU
m-m
am
a-K
lww
lnklworn.
mr-
tvmw
oum
..
..
..
..
..
..
..-.
..
-,-.-.-.--m-~mmm.~m-mm-y-~--~
000000000000000000000
000000000000000000000
+++++++++
++++++++++
++
wu
wu
uw
wu
wu
uw
uu
ww
uw
uu
~mmwmn.++oo-mmammqvoqb
aN-vm.waOw.mmmmwm”mb~
mwommmm.m+mmqaoyqq~mm
fi.mm
mqbm
omm
o-timm
m~w
mo
..
..
..
..
..
..
.....
--.---.Nm
-.Nm
mm
mN
m””{&
...,-??0
0,,
Luww
u-cfm
b-C
OW
*m
ww
r.eJ
mo-
..
..
mmvm
+IJJ0mC
(
&
------????W
luww
owa
-C
4Ulom
mm
at.
.-,-ofJ)
------????U
wuw
I.m
-mpumm
mam
mr-m
u-lo
.,m
“.ilnlc!w
00.
yy?
U.IU
LUwwm
vvm.-*acorn
..
..--m
m.wmmwmcocommmmm
------------------
10
!,,,,,
,,,,,,
,W
ww
ww
uu
.ruu
ulw
wlu
zwvvm
moucxl.
m-lm
-a-mm
.-m-o
wm
amm
>um
mm
inw
mm
om
qt.fm
v~zw
mo.-w
wom
mm
~m
w-1
mwmvm.
mbmt.
+mm
/
Oooooooooooom
Wooooooooo
0000
C1.0
00ooooooooou
lzem
c4m
owcoc+
mow
m-
-.(-4mm
@)m
*~~m
**:***W*Q.
*+:3*E**U*-I*W*LE
Z*O**C
-4
:*.*0*z**L*Z*3*n
59
%0.ml-l-1-100z0n1-
a-13n.
0LIn!?
xl-lV
Kauu
l-t-zw
-!m0E-la-1W
Evu
0En
60
gggggoooooooooooo’”-mooooooooooooo~
o@J
+++++++++
+++++++++-+
-w
wm
ww
uw
uw
wu
ww
ww
~w~+w
+N
mom
FN
mm
F*m
-v~-~
~~
o~o
omm
*ww
Nm
w~
wm
-*+w
mm
oOo
mm
mm
mm
om--w
mt-~
--mw
~o-
r--mm
.O0m
t--*~M
mW
OO
~O
~8
..-:-,
-------.
..
..
..
..
..
..
bb
m+m
mm
mm
m+m
mm
mm
m--+-
pn~W
ww
watw
mw
maw
mm
omr-
W-w
m.
..
.v-----
0yu.24
t-ill
0000
yyy~
UJW
UU
.Ju
l.xo-
oomw
mom
t-w
mm
m.
..
..=:
@J- ,--
00000
0000
c) 8++++++
wuwuUJw
bwmwt-~
wmwomci
Fwomlnw
mo@4@YC’4-
......
-NWCNNC4
..
..
c41’9
*I-
‘F-
‘--------
”--’---8
88
000000000000000000
11111111
IIllllolil
+++
uuuuwwuwwuwuUwwww~wUw
ww-mwawtio+mw++m.-~mmam
bwtimmwNmmmO-*+Ow@o~mm
mmow-ww-moomo+m-~wm~-
mtimmmmwbmw-wmmab+bmaw
..-,
..
..
..
..
..
..
..
..
..
+m
mm
wom
wm
***mm
wW
~w
--m
oo-yy?
Wlu
ul
mm
om
tgm000m
cuo
..
.
m-m
titwtim
m-tw
-mm
m00000000000
1ss11111111
UJu
lulu
ulw
lulu
uu
wm
um-O
aOam
O-
m.-m
ww
mm
m-+
wm
mm
t-vbm
-wO
a3
mcom
--.--vbaa
..
..
..
..
..
.m
t-4.-m.au
)r4Fm
-
wm
w~
-0000
:,,1
1W
lul.u
uu
wm
omm
In-u
-l+g
r-Nm
’-r-w
-m+
..
..
..
.
A
Lu
tw
=a-C
ixzw
ow>a
-=x
r??+1-
Wi$Ii
v.><W
I-4&
zzl-
>X
z4(J
Uam
ct-
1-
n
X:Z
31(Y
-lffu
lL
I+ZUlw
3
KS
>W
I-(JI
❑xw
=(9
LU
31+Z
Zgw
U)>
Zi-w
0s0l-lc!ll-In
.ul
I-lW
i+Jzzx
J:*Kw0.
oooooooooooo8~S
88G$Z:
000000000000
+++++++++-‘+++++++++-+-
IL!uu
wwwuIllww
wLuw
UJw
mw
w+
w+
mm
omr-m
mm
Fu
m-w
N-m
mN
O-
Om
mw
um
mm
w-w
m-r-k
mm
mO
ffl 8m
min
mm
mom
--mm
t-m-m
meJ
Om
Oti-m
m.
oomt-.
t-m-m
mm
wO
omo
..
..
..
..
..
..
..
..
~;;U
UIU
JO
WN
mam
mr.
mm
mm
..
.s-.-m
-~aw
m00000
11118W
WW
LU
Wt-m
mc-+
-m
mm
wm
W!n
t-oobw
bwm
,..
,.
.-w
N-m
Com
qem
00000
11111W
ww
uu
oc+m
ombt-tiw
ln-om
mr-
Fo
ww
CD
..
..
..-
.-*.-O
W19W
*I+00000
11111U
uw
uiw
m~-w
mw
oN~m
F-*----
emm
mo
..
..
.m
-t-to-
Ow
.-.
ILL
(7)!.w
-t--
I-lw.
.-N
..
..
------
013::$’$W
Wu
uu
lluC
iWti
o-m
~g;n$;.
..
..
...-,-
cq..
4m
m1-
Zm
00
.m.-1-10(J
wz
m0
m.i-
m<-13n0n
-1a1-
01-
-1m0Kn.
0!
00Ka.
i-ElS090”Z
=d
NPS100020003000400050006000700080009000
1000011000i 20001300014000150001600017000180001900020000
TALLY 5MEAN
9.39391E-208.55653E-206.67066E-207.04297E-206.79854E-206.40329E-206.30279E-206.18213E-206.37404E-206.19940E-206.41550E-206.66364E-206.40622E-206.24940E-206.14f67E-206.18275E-205.98841E-206.01460E-205.96264E-205.90955E-20
ERROR.3471.2516.2251.1841.1649.1498
1355;1272.1198.1148.1081
1112:1077.io4i
1012:0971.0953.0915.0894.0894
FOM6972687476798584818282717172727373747370
**************************************************************************DUMP NO. 2 ON FILE RUNTPF NPS = 20000 CTM = 1.77
NPS100020003000400050006000700080009000
1000oIlooo12000130001400015000160001700018000190002oooo
TALLY 5MEAN
9.30107E-208.47200E-206.60475E-206.97337E-206.73137E-206.34001E-206.24051E-206.?2105E-206.31105E-206.13814E-206.35210E-206.59779E-206.34292E-206.18765E-206.08098E-206.12165E-205.92923E-205.95516E-205.9037fE-205.85115E-20
ERROR.3471.2516.2251.1841.1649.1498
1355:1272.1198
1148:1081
1112:1077
to41:1012.0971.0953.0915.0894.0894
Fot.1 THE DIFFERENCE BETWEEN THESE TWO IS PROBABLY70 STATISTICALLYSIGNIFICANT,BUT MCNPDOESNOT72 DIRECTLY GIVE THE ERRORINTHE DIFFERENCE.68 HOWEVER, THIS CAN BE FIGURED OUT FROM THEFOM74 CHARTS BECAUSE WE CAN DO BATcH STATISTICS,7679
HERE 20 BATCHES OF 1000.
8584 AVERAGE SCORE DUE TO PARTICLES 9001-10,000818271S 1O,OOO*6.13S14E-2O -9000* 6.31105E-2082717172727373747370
1000
**************************************************************************DUMP NO. 2 ON FILE RUNTPE NPS = 2COO0 CTM = 1.77
FIG. 46. PROCESS FOR OBTAINING BATCH STATISTICS.
Fig. 46. Processforobtainingbatchstatistics.
61
1 I CELL21 DENSITIES 1% DIFFERENT
mN
Ip= 2.03E-49.301 070E-207.642930E-202.870250E-208.079230E-205.763370E-204. 3&i27 CIE-205.64351OE-2O5.284830E-207.83105OE-2O4.581950E-208.491700E-209.300380E-203.284480E-204.169140E-204.587600E-206.731700E-202.850510E-206.395970E-2o4.977610E-204.85251oE-2o
I (CELL21 IS THIN FORCEDCOLLISIONP=2.05~3E-4
CELLNEARRINGDETECTOR)9.393910E-207.719150E-2CJ2.898920E-208.159900E-205.820820E-204.427040E-205.699790E-205.337510E-207.909320E-204.627640E-20 20 MEANS FC)RBATCHESOF 1000 HISTORIES FROMFOM TABLES8.5’j’6500E-209.393180E-203.317180E-204.210740E-204.633450E-206.798950E-202.878970E-206.459830E-205.027360E-204.900840E-20
RELERROR=7.783477E-02 ~
\
HIGHER DENSTIY ISGOhJTRIBIJTING ABOUT
1% MORE THAN LOWER DENSITY
MEAN FOR
P’=2.0503E-4 CONCLUSION:
MEANOFDIFFERENCES~ MEAN-5.840000E-22
MEAN FORP=2.03E-4
5.85115E-20 0.0894 5.90955E-20 0.0894 USING CORRELATEDSAMPLING,HAVE FOUNDA 1% DIFFERENCETOWITHIN 8%DESPITET~~OlhFRRQf+!N EOTHCALCULAT!ONS*.- . . .. .
Fig.47. Meansforstandarddensityand perturbeddensityproblems.
XXI. SUMMARY
SUMMARYPROBLEM#1
Page Particles F1 F4 F5This Time am O*Report Techniques Part/Min ;6M FOM FOM Comments
6
8
11
12
14
17
19
25
26
Analog No particlesget pastcell 14 (Point detectorcontributions only fromcell 21).
&s.urnes particles below.01 MeV do not contn-bute; No particlesbeyond cell 13.
Particles now penetrat-ing concrete; use “tracksentering” to refineimportances.
Keep refined splittingon “tracks entering”information.
Factor of 2 gained byenergy roulette.
Implicit capture andweight cutoff did reducethe history variance, buttime per historyincreased too much.Thus analog capturebetter.
Forced collision allowsthe point detector (F5)to get tallies.Material too thin.
DXTRAN successfid fortallies 4 and 5, but tooslow; angle biasingdefinitely helps, workon speed.
DXCPN solves sueed
3919 00.61 min6425
0 0
Energy Cutoff.01 MeV
13968 00.60 min23280
0 0
Geometry Splitting(factor of 2,cells 2-19)Energy Cutoff
2118 5.87E-70.60min 0.243530 27
0 0
Refined SplittingEnergy Cutoff
15200.58 min2620
5.03E-70.2723
7.21E-14 O1.001
Energy RouletteRefined SplittingEnergy Cutoff
46990.61min7703
8.38E-70.1850
1.92E-13 O0.644
Weight Cutoff/Implicit CaptureEnergy RouletteRefined SplittingEnergy Cutoff
20990.61min3441
5.62E-70.1937
5.59E-14 O0.73 02
Forced CollisionEnergy RouletteRefined SplittingEnergy Cutoff
31617 5.59E74.61min 0.0686858 45
7.53E-14 2.61E-170.27 0.292 2
DXTRANForced CollisionEnergy RouletteRefined SplittingEnergy Cutoff
1.24E-13 7.62E-170.21 0.2215 14
2231 7.35E-71.43min 0.231560 12
DXCPNand 11427 7.32E-7 1.22E-13 7.21E-17DXTRANForced CollisionEnergyrouletteRefined SplittingEnergvCutoff
2.60min 0.10 0.095 0.104395 34 42 34
problem.Note F1 tally4395/1560=34/12=FOMratio.
63
PROBLEM#1 (continued)
Page Particles FI F4 F5This Time am Omr cmReport Techniques Part/Min FOM FOM FOM Comments
27 Ring DetectorEnergyRouletteDXTRAN/DXCPNForced CollisionRefinedSplittingEnergyCutoff
30 Cone BiasingRing DetectorDXTRAN/DXCPNForced CollisionRefined SplittingEnergyCutoffEnergyRoulette
32 ExponentialBiasRing DetectorDXTRAN/DXCPNForced Coll~sionRefinedSphttingEnergyRouletteEnerm Cutoff
117552.60min4521
60492.60 min2327
54042.59 min2086
6.54E-70.1038
6.82E-70.1132
6.79E-70.1133
1.16E-130.09344
1.22E-130.0924.5
1.05E-130.09839
6.78E-170.09641
7.37E-170.09740
6.43E-170.1035
Ring detector looksmarginallybetter.
Conebias has littleeffectbecause—~sourceparticlesdiequickly.Remove conebias below.
Exponentialsourcebiaslooksmarginallydetri-mental.
SUMMARY PROBLEM#2NewSource95Y0at 2 MeV and5V0at 14 MeV
Page Particles F1 F4 F5This Time o~ O*Report Techniques Part/Min ;~M FOM FOM Comments
33 Splitting(same) 33092EnergyCutoff,Ring 4.60minDetector/Forced 7194Collision/DXTRAN/DXCPN(subsequentruns use above tech-niquesunlessspeci-fied otherwise)
35 SourceEnergyBias 63064.63min1362
36 No sourceenergy 66475bias, energy 4.61 minroulette 14420
39 SourceEnergyBias 16957EnergyRoulette 4.61 min
3678
42 Turnoff splitting 46770anduse importances4.60 minasweightwindow 10167
4.43E-80.234
4.90E-80.1116
6.22E-80.159
5.04E-80.08033
5.87E-80.273
7.57E-150.224
8.61E-150.1117
9.80E-150.1410
8.81E-150.7637
1.05E-140.243
4.35E-180.214
5.1OE-180.1117
5.94E-180.159
5.14E-180.07638
5.82E-180.233
Problem much harderbecausesourcespectrummuch softer.Factor15lesstransmission.
Factor of fourimprovement by E bias.
Worsethan sourceenergybias, better than noenergydiscrimination.
Good idea to use both inthis problem.
Within statistics, aboutthe same as splitting.
65
SubsequentRunsUse WeightWindowUnlessOtherwiseSpecified
Page Particles FI F4 I%This Time %.r ~m anReport Techniques Part/Min FOM IFOM FOM Comments45
48
50
51
53
55
56
59
Windowfrom 46770importance 4.79mingeneratoron 9764(spatial)
5.87E-80.272
1.05E-80.243
5.82E-180.233
Note this run and pre-vious run tracked;onlydifferenceis a 4Y0reduction in speed.
DXTRAN turned offwhilewindowis beingoptimizedforpenetra-tion.
Use generatedspace 28144window,turn 4.63DXTRAN Off, 6079space-energygeneratoron
3.66E-80.196
5.08E-150.660
1.96E-180.760
4.24E-180.264
Space-energywindowgivesdramatic improve-ment.
Note goodimprovementwith sourceenergybias.
Space-energywindow79266generatedabove; 4.61minDXTRANOff 17194
4.48E-80.07043
8.24E-150.234
Sourceenergybias 71167so that particles 4.61minstart within 15438space-energywindow
4.92E-80.05475
8.48E-150.292
9.20E-180.510
Sameas above, 74051exceptcorrectbad 4.60minwindow 16098
5.09E-80.5572
8.61E-150.28
‘2
2.20E-180.312
Murphy’sLaw.
Exponentialtransformworkswellwith weightwindow.
Exponentialtrans- 81021form, space-energy 4.60minwindow,source- 17613energybias
4.85E-80.041126
1.06E-140.159
4.62E-180.168
8.59E-2010
Exponentialtransformrequireswindow.
Same as above, 90897exceptremove 4..60minwindow 19760
4.05E-80.351
1.90E-1610
8.46E-15 4.86E-18GRANDFINALE 51909 4.74E-8Turn DXTRANback 4.60min 0.053 0.055 0.054on 11285 77 71 73
REFERENCES
1. Los AlamosMonte Carlo Group, “MCNP-A Gen-eral Monte Carlo Code for Neutron and PhotonTransport,” Los AlamosNational Laboratory reportLA-7396-M(Rev.) (April 1981).
2. Thomas E. Booth, “Monte Carlo Variance Com-parison for Expected Value Versus Sampled Split-ting,” NuclearScienceand Engineering89, 305-309(1985).
66
APPENDIX
InputFileDifferencesforMCNP Version3A
The calculationsdescribed in this report were donewith MCNP version 2D, and some of the input filespecificationshave been changed in version 3A. Thisappendixwas added to aid the reader who wants to runthe sampleproblem on MCNP3A.
The source specification(cards SRCI, S1,and SP ofFig.2) willhave to be altered substantially.In addition,the reader shouldbe aware of the followingchanges:
1. Particle Types:
MCNP3A will recognizetwo particle types with thefollowingmnemonics:
N = neutronP = photon
2. DataCards:
The particletype of each data cardwill be the firstdataentryand no longerappearas partof the datacardname.Thismeansthatthe followingdatacardsarerenamed:
New OldName Name(s) Description
IMPCUT
PHYS
WWGE
WWP
ESPLTEXTDXT
FCLDXC
IN,IPCUTN,CUTP
ERGN,ERGPWFN,WFPWFN,WFPWGEN,WGEP
WDWN,WDWP
NSPLT,PSPLTEXTYN,EXTYPDXN,DXP
FCN,FCPDXCPN,DXCPP
importancetime, energy, weight
cutoffsenergy physics cutoffsweight window boundsweight window energiesweight window
generator energiesweight window game
parametersenergy splitting/rouletteexponential transformDXTRAN sphere
specificationforced collisionsDXTRAN cellcontributions
3.
4.
The new root entry will appear in columns 1-5;theN or P data type will be the first entry beyondcolumn 5. If the first data entry is not an N or P,therewillbe a fatalerror. Note that onlyone particletype may be specified. If the particle type is in-consistent with the problem mode, there will be awarning error. A warning rather than a fatal errorwillbe issuedso that a coupledneutron/photon runmay be switched to a neutron-only run withoutremoving all the photon data cards. In MCNP3Athe old data cards will be accepted with a warningthat they willbe obsoletein MCNP3B.
MODE Card:
The MODE card will specifi the problem particletypes.Examples:
MODE N (old mode O)MODE N P (old mode 1)MODE P N (old mode 1)MODE P (old mode 2)
If both N and P are specified, the order does notmatter for MCNP3A and the two entries must beseparated by at least one space. The space is re-quired so that future versions of the code can haveparticle types with more than one charactermnemonics.
The old MODE card will be accepted with a warn-ing that it willhave differententries in MCNP3B.
TallyParticleTypes:
Whether a tally is a neutron or photon tally isspecifiedby an N or P as the first entry on the tallyFn card regardless of the tally number. ForMCNP3A, if the N or P is missing then a warningwillbe issued and the particle type willbe assumedfrom the tally number as in previous versions.Examples:
F4 P cl C2 C3 photon flux tallyF15 N x y z ro neutron detector tallyF7 cl C2 C3 neutron heating tally:
warning issued.
67
The neutron and photon heating tallies may be a fatal error if it contains anything more thanadded togetherby having both an N and a P as the constants.Examplesof proper usage:firstand seconddata entries.The N and P may be inany order, i.e., P and NYand they must be separated F6 P N c1 C2 C3bv a snace.The F6 and’F16 tally types are the only FM6 cl. .tally types that may be added in ‘thisway. A cor-respondingFMn card for the combined tally causes
68
F36 N P c1 C2 C3 C4FM36 (Cl) (C2) (C3) (C4) (C5)
*U.S. GOVERNMENT PRINTING OFFICE: 1989-673-034/00184
This report has been reproduced directly from thebest available copy.
It is available to DOE and DOE contractors from theOffice of Scientific and Technical Information,P.O. Box 62,Oak Ridge, TN 37831.Prices are available from(615) 576-8401.
It is available to the public from theNational Technical Information Service,US Department of Commerce,5285 Port Royal Rd.Springfield, VA 22161.
LosN A T I O N A L L A B O R A T O R Y
AlamosLos Alamos, New Mexico 87545
